Perspective Chapter: Families of Seventh-Order KdV Equations Having Traveling Wave and Soliton Solutions

Abstract

In this paper, we consider the problem of finding traveling wave solutions to the generalized seventh-order KdV equation (KdV7). Solitons are non-linear waves that exhibit extremely unexpected and interesting behavior—solitary waves that propagate without deformation. We use different approaches in order to find one and multisoliton solutions. Soliton travels through liquid, solid, and gaseous media and even as electron waves through an electromagnetic field. Making use of a traveling wave transformation, we obtain a non-linear ode, which is solved using either hyperbolic or elliptic algorithm. We also use the Hirota method to get the bilinear form, and then we may obtain multisoliton solutions. In the end, we consider the forced KdV7.

1. Introduction

One of the most notable achievements in the second half of the twentieth century, which also clearly illustrates the underlying unity of Mathematics and Nonlinear Physics, is the Theory of Solitons. Solitons are nonlinear waves exhibiting extremely unexpected and interesting behavior—solitary waves propagating without deformation.

The other waves, the nonlinear ones, are less familiar and are very different from the linear ones. A wave in the sea approaching the shore is a good example of a nonlinear wave. Note that the amplitude, wavelength, and speed vary as the wave advances, while in linear waves, these are constant. The distance between the crests decreases, the height of the waves increases as they perceive the bottom, and the speed changes. The upper part of the wave overtakes the lower part, falls on it, and the wave breaks. There are even more intricate phenomena such as two waves that intersect, interact in complicated and nonlinear ways, and give rise to three waves instead of two.

Now we come to solitons. During a horseback ride around Edinburgh, on the Union Canal in Hermiston, very close to the Riccarton campus of Heriot-Watt University, the Scottish engineer John Scott-Russell watched as a barge was towed along a narrow canal by two horses that pulled from land to obtain a more efficient design of boats.

A decisive step in the theory of integrable systems was the integration of the KdV equation. Thus, Gardner, Greene, Kruskal, and Miura observed that if we consider a potential u(x) for the stationary Schrödinger equation on the line, the corresponding scattering data are transformed extremely easily when the potential changes as long as u(x, t) satisfies the KdV equation. Therefore, given an initial condition u(x) for KdV, we can find the associated scattering data and determine its evolution immediately.

In this paper, we consider the following generalized seventh-order KdV equation (KdV7 for short):

$$u_t+{\mathit{au}}^3{u}_x+{\mathit{bu}}_x^3+{\mathit{cuu}}_x{u}_{2x}+{\mathit{du}}^2{u}_{3x}+\alpha u_{2x}{u}_{3x}+\beta u_x{u}_{4x}+\gamma {\mathit{uu}}_{5x}+u_{7x}=0.$$E1

This nonlinear evolution equation describes the behavior of physical phenomena such as shallow water waves and plasmas. Its conservation laws were determined to predict its complete integrability [1, 2]. In Ref. [3], Wazwaz obtained one and two soliton solutions for the following special cases:

• The seventh-order Sawada-Kotera-Ito equation:

  $$u_t+252u^3{u}_x+63u_x^3+378u_x{u}_{2x}+126u^2{u}_{3x}+63u_{2x}{u}_{3x}+42u_x{u}_{4x}+21{\mathit{uu}}_{5x}+u_{7x}=0.$$E2

• The seventh-order Lax equation:

  $$u_t+140u^3{u}_x+70u_x^3+280u_x{u}_{2x}+70u^2{u}_{3x}+70u_{2x}{u}_{3x}+42u_x{u}_{4x}+14{\mathit{uu}}_{5x}+u_{7x}=0.$$E3

• The seventh-order Kaup-Kuperschmidt equation

$$u_t+2016u^3{u}_x+630u_x^3+2268u_x{u}_{2x}+504u^2{u}_{3x}+252u_{2x}{u}_{3x}+147u_x{u}_{4x}+42{\mathit{uu}}_{5x}+u_{7x}=0.$$E4

These three cases of the seventh-order KdV equation are completely integrable. This means that each of these equations admits an infinite number of conservation laws, and as a result, each gives rise to N-soliton solutions. We aim to describe the families of these KdV7 that admit soliton and cnoidal wave solutions.

2. Cnoidal wave solutions

Let

$$u\left(x,t\right)=p+q\wp \left(x-\mathit{\lambda t}+{\xi }_0;g_2,g_3\right).$$E5

Then

$$\begin{array}{l}{u}_t+{\mathit{au}}^3{u}_x+{\mathit{bu}}_x^3+{\mathit{cuu}}_x{u}_{2x}+{\mathit{du}}^2{u}_{3x}+\alpha u_{2x}{u}_{3x}+\beta u_x{u}_{4x}+\gamma {\mathit{uu}}_{5x}+u_{7x}=\\ \frac{1}{2}q\sqrt{-g_2\wp -g_3+4{\wp }^3}[2{\mathit{ap}}^3-2{\mathit{bg}}_3{q}^2-{\mathit{cg}}_2\mathit{pq}-36\gamma g_{2}p-24\beta g_{3}q-1440g_3-2\lambda \\ +\left(6{\mathit{ap}}^{2}q-2{\mathit{bg}}_2{q}^2-{\mathit{cg}}_2{q}^2+24{\mathit{dp}}^2-12\alpha g_{2}q-36\beta g_{2}q-36\gamma g_{2}q-4032g_2\right)\wp +\\ 6p\left({\mathit{aq}}^2+2\mathit{cq}+120\gamma +8\mathit{dq}\right){\wp }^2+\\ 2\left({\mathit{aq}}^3+4{\mathit{bq}}^2+6{\mathit{cq}}^2+12{\mathit{dq}}^2+72\mathit{\alpha q}+120\mathit{\beta q}+360\mathit{\gamma q}+20160\right){\wp }^3],\end{array}$$E5a

where $\wp =\wp \left(x-\mathit{\lambda t}+{\xi }_0;g_2,g_3\right).$

The system to be solved is

$$\begin{array}{l}2{\mathit{ap}}^3-2{\mathit{bg}}_3{q}^2-{\mathit{cg}}_2\mathit{pq}-36\gamma g_2-24\beta g_{3}q-1440g_3+2\lambda =0.\\ 6{\mathit{ap}}^{2}q-2{\mathit{bg}}_2{q}^2-{\mathit{cg}}_2{q}^2+24{\mathit{dp}}^2-12\alpha g_{2}q-36\beta g_{2}q-4032g_2=0.\\ 6{\mathit{apq}}^2+12\mathit{cpq}+720\gamma +48\mathit{dpq}=0.\\ 2{\mathit{aq}}^3+8{\mathit{bq}}^2+12{\mathit{cq}}^2+24{\mathit{dq}}^2+144\mathit{\alpha q}+240\mathit{\beta q}+40320=0.\end{array}$$E6

We will have a solution for the following parameter values:

$$\begin{array}{l}{g}_2=\frac{6p^2\left(\mathit{aq}+4d\right)}{2{\mathit{bq}}^2+{\mathit{cq}}^2+12\mathit{\alpha q}+36\mathit{\beta q}+4032}.\\ g_3=\frac{2{\mathit{ap}}^3-{\mathit{cg}}_2\mathit{pq}-36\gamma g_2+2\lambda }{2\left({\mathit{bq}}^2+12\mathit{\beta q}+720\right)}.\\ p=-\frac{120\gamma }{q\left(\mathit{aq}+2c+8d\right)}.\\ {\mathit{aq}}^3+2\left(2b+3c+6d\right)q^2+24\left(3\alpha +5\beta \right)q+20160=0.\end{array}$$E7

Example. Let

$$\begin{array}{l}{u}^{\left(0,1\right)}\left(x,t\right)-63.5567u{\left(x,t\right)}^3{u}^{\left(1,0\right)}\left(x,t\right)-5089.7u^{\left(1,0\right)}{\left(x,t\right)}^3+0.939611u\left(x,t\right)u^{\left(1,0\right)}\left(x,t\right)u^{\left(2,0\right)}\left(x,t\right)\\ +0.471886u{\left(x,t\right)}^2{u}^{\left(3,0\right)}\left(x,t\right)+0.0310296u^{\left(2,0\right)}\left(x,t\right)u^{\left(3,0\right)}\left(x,t\right)+0.626069u^{\left(1,0\right)}\left(x,t\right)u^{\left(4,0\right)}\left(x,t\right)\\ +0.48252u\left(x,t\right)u^{\left(5,0\right)}\left(x,t\right)+u^{\left(7,0\right)}\left(x,t\right)=0.\end{array}$$E8

See Figure 1.

Figure 1.

$u\left(x,t\right)=0.480062+\wp \left(-7.21261t-x-\left(4.70653-5.33465i\right);0.0139651,-0.0000274235\right)$.

Let now

$$u\left(x,t\right)=B+C{\mathrm{cn}}^2\left(\mathit{\omega x}-\mathit{\lambda t}+{\xi }_0,m\right).$$E9

Then

$$\begin{array}{l}{u}_t+{\mathit{au}}^3{u}_x+{\mathit{bu}}_x^3+{\mathit{cuu}}_x{u}_{2x}+{\mathit{du}}^2{u}_{3x}+\alpha u_{2x}{u}_{3x}+\beta u_x{u}_{4x}+\gamma {\mathit{uu}}_{5x}+u_{7x}=\\ 2C\mathrm{cn}\mathrm{sn}\mathrm{dn}[-\left(\begin{array}{l}\lambda -{\mathit{aB}}^3\omega -2\mathit{BcC}{\omega }^3+4B^{2}d{\omega }^3+2\mathit{\text{BcCm}}{\omega }^3-8B^2\mathit{dm}{\omega }^3+8C{\mathit{\alpha \omega }}^5-24\mathit{Cm}{\mathit{\alpha \omega }}^5+16{\mathit{Cm}}^2{\mathit{\alpha \omega }}^5\\ +8C{\mathit{\beta \omega }}^5-24\mathit{Cm}{\mathit{\beta \omega }}^5+16{\mathit{Cm}}^2{\mathit{\beta \omega }}^5-16B{\mathit{\gamma \omega }}^5+136\mathit{Bm}{\mathit{\gamma \omega }}^5-136{\mathit{Bm}}^2{\mathit{\gamma \omega }}^5+64{\omega }^7\\ -2112m{\omega }^7+5952m^2{\omega }^7-3968m^3{\omega }^7\end{array}\right)\mathrm{cn}\\ +\omega \left(\begin{array}{c}3{\mathit{aB}}^{2}C-4\mathit{BcC}{\omega }^2+4{\mathit{bC}}^2{\omega }^2+2{\mathit{cC}}^2{\omega }^2-8\mathit{BCd}{\omega }^2+8\mathit{\text{BcCm}}{\omega }^2-4{\mathit{bC}}^{2}m{\omega }^2-2{\mathit{cC}}^{2}m{\omega }^2-12B^2\mathit{dm}{\omega }^2\\ +16\mathit{\text{BCdm}}{\omega }^2+16C{\mathit{\alpha \omega }}^4-88\mathit{Cm}{\mathit{\alpha \omega }}^4+88{\mathit{Cm}}^2{\mathit{\alpha \omega }}^4+16C{\mathit{\beta \omega }}^4-136\mathit{Cm}{\mathit{\beta \omega }}^4+136{\mathit{Cm}}^2{\mathit{\beta \omega }}^4+16C{\mathit{\gamma \omega }}^4\\ +240\mathit{Bm}{\mathit{\gamma \omega }}^4-136\mathit{Cm}{\mathit{\gamma \omega }}^4-480{\mathit{Bm}}^2{\mathit{\gamma \omega }}^4+136{\mathit{Cm}}^2{\mathit{\gamma \omega }}^4-4032m{\omega }^6+24192m^2{\omega }^6-24192m^3{\omega }^6\end{array}\right){\mathrm{cn}}^3\\ +\omega \left(\begin{array}{l}3{\mathit{aBC}}^2-4{\mathit{bC}}^2{\omega }^2-4{\mathit{cC}}^2{\omega }^2-4C^{2}d{\omega }^2-6\mathit{\text{BcCm}}{\omega }^2+8{\mathit{bC}}^{2}m{\omega }^2+8{\mathit{cC}}^{2}m{\omega }^2-24\mathit{\text{BCdm}}{\omega }^2\\ +8C^2\mathit{dm}{\omega }^2+72\mathit{Cm}{\mathit{\alpha \omega }}^4-144{\mathit{Cm}}^2{\mathit{\alpha \omega }}^4+120\mathit{Cm}{\mathit{\beta \omega }}^4-240{\mathit{Cm}}^2{\mathit{\beta \omega }}^4+240\mathit{Cm}{\mathit{\gamma \omega }}^4+360{\mathit{Bm}}^2{\mathit{\gamma \omega }}^4\\ -480{\mathit{Cm}}^2{\mathit{\gamma \omega }}^4-20160m^2{\omega }^6+40320m^3{\omega }^6\end{array}\right){\mathrm{cn}}^5\\ +\omega \left({\mathit{aC}}^3-4{\mathit{bC}}^{2}m{\omega }^2-6{\mathit{cC}}^{2}m{\omega }^2-12C^2\mathit{dm}{\omega }^2+72{\mathit{Cm}}^2{\mathit{\alpha \omega }}^4+120{\mathit{Cm}}^2{\mathit{\beta \omega }}^4+360{\mathit{Cm}}^2{\mathit{\gamma \omega }}^4-20160m^3{\omega }^6\right){\mathrm{cn}}^7].\end{array}$$

Next, we equate to zero the coefficients of ${\mathrm{cn}}^j$ to obtain an algebraic system of nonlinear equations. This system admits a solution under the condition

$${\mathrm{\Delta }}_1{\mathrm{\Delta }}_2=0,$$E10

where

$$\begin{array}{l}{\mathrm{\Delta }}_1=17640a^2+27a{\alpha }^2\gamma +90\mathit{\text{aαβγ}}+180a{\mathit{\alpha \gamma }}^2+75a{\beta }^2\gamma +300a{\mathit{\beta \gamma }}^2-840\mathit{ab\gamma }+300a{\gamma }^3-126\mathit{a\alpha c}-210\mathit{a\beta c}\\ -1260\mathit{a\gamma c}-504\mathit{a\alpha d}-840\mathit{a\beta d}-2520\mathit{a\gamma d}+10b^2{\gamma }^2+14{\mathit{bc}}^2-3\mathit{\text{αbγc}}-5\mathit{\text{bβγc}}+10b{\gamma }^{2}c+112\mathit{bcd}+224{\mathit{bd}}^2\\ -12\mathit{\text{αbγd}}-20\mathit{\text{bβγd}}-20b{\gamma }^{2}d+14c^3-3\mathit{\alpha \gamma }{c}^2-5\mathit{\beta \gamma }{c}^2+126c^{2}d+336{\mathit{cd}}^2-15\mathit{\text{αγcd}}-25\mathit{\text{βγcd}}-30{\gamma }^2\mathit{cd}+224d^3\\ -12\mathit{\alpha \gamma }{d}^2-20\mathit{\beta \gamma }{d}^2-30{\gamma }^2{d}^2,\text{and}\\ {\mathrm{\Delta }}_2=28224a^2+3a{\alpha }^3+3a{\alpha }^2\beta +63a{\alpha }^2\gamma -63a{\mathit{\alpha \beta }}^2+234\mathit{\text{aαβγ}}+297a{\mathit{\alpha \gamma }}^2-135a{\beta }^3+135a{\beta }^2\gamma +168\mathit{a\alpha b}\\ +3192\mathit{a\beta b}+675a{\mathit{\beta \gamma }}^2-3528\mathit{ab\gamma }+405a{\gamma }^3-252\mathit{a\alpha c}+588\mathit{a\beta c}-2772\mathit{ac\gamma }-1008\mathit{a\alpha d}-3024\mathit{a\beta d}-3024\mathit{a\gamma d}\\ +504b^3+4{\alpha }^2{b}^2-16\mathit{\alpha \beta }{b}^2+84\alpha b^2\gamma -20{\beta }^2{b}^2+-60\beta b^2\gamma +360b^2{\gamma }^2+84b^{2}c-2016b^{2}d-70{\mathit{bc}}^2+2{\alpha }^2\mathit{bc}\\ -12\mathit{\text{αβbc}}+48\mathit{\text{αbcγ}}+10{\beta }^2\mathit{bc}-120\mathit{\text{βbcγ}}+270\mathit{bc}{\gamma }^2-672\mathit{bcd}+2016{\mathit{bd}}^2-6{\alpha }^2\mathit{bd}+12\mathit{\text{αβbd}}-108\mathit{\text{αbγd}}\\ +90{\beta }^2\mathit{bd}-180\mathit{\text{βbγd}}-270b{\gamma }^{2}d+7c^3-2\mathit{\alpha \beta }{c}^2+3\alpha c^2\gamma +10{\beta }^2{c}^2-45\beta c^2\gamma +45c^2{\gamma }^2+168c^{2}d+1008{\mathit{cd}}^2\\ -3{\alpha }^2\mathit{cd}+6\mathit{\text{αβcd}}-54\mathit{\text{αcγd}}+45{\beta }^2\mathit{cd}-90\mathit{\text{βcγd}}-135c{\gamma }^{2}d.\end{array}$$

The Eqs. (2)–(4) obey the condition in Eq. (10).

Solving the system we obtain the solutions as follows:

$$\begin{array}{l}\lambda =\omega \left(\begin{array}{c}{\mathit{aB}}^3+8B^2\mathit{dm}{\omega }^2-4B^{2}d{\omega }^2-2\mathit{\text{BcCm}}{\omega }^2+2\mathit{BcC}{\omega }^2\\ +16B{\mathit{\gamma \omega }}^4+136\mathit{B\gamma }{m}^2{\omega }^4-136\mathit{B\gamma m}{\omega }^4-8\mathit{\alpha C}{\omega }^4\\ -8\mathit{\beta C}{\omega }^4-16\alpha {\mathit{Cm}}^2{\omega }^4-16\beta {\mathit{Cm}}^2{\omega }^4+24\mathit{\alpha Cm}{\omega }^4\\ +24\mathit{\beta Cm}{\omega }^4+3968m^3{\omega }^6-5952m^2{\omega }^6+2112m{\omega }^6-64{\omega }^6\end{array}\right).\\ \begin{array}{ll}B=& -\frac{4\left(2m-1\right){\omega }^2}{3\left({\mathit{aC}}^2-2\mathit{cCm}{\omega }^2-8\mathit{Cdm}{\omega }^2+120\gamma m^2{\omega }^4\right)}({\mathit{bC}}^2+{\mathit{cC}}^2+C^{2}d-18\mathit{\alpha Cm}{\omega }^2-30\mathit{\beta Cm}{\omega }^2\\ & -60\mathit{\gamma Cm}{\omega }^2+5040m^2{\omega }^4).\end{array}\\ \begin{array}{ll}{\mathit{aC}}^3+& \left(-4\mathit{bm}{\omega }^2-6\mathit{cm}{\omega }^2-12\mathit{dm}{\omega }^2\right)C^2+\left(72\alpha m^2{\omega }^4+120\beta m^2{\omega }^4+360\gamma m^2{\omega }^4\right)C\\ & -20160m^3{\omega }^6=0.\end{array}\end{array}$$E11

• Sawada-Kotera-Ito Eq. (2):

$$\begin{array}{ll}C=& 2m{\omega }^2\\ \lambda =& 4\omega (63B^3+252B^{2}m{\omega }^2-126B^2{\omega }^2+336{\mathit{Bm}}^2{\omega }^4-336\mathit{Bm}{\omega }^4+84B{\omega }^4\\ & +152m^3{\omega }^6-228m^2{\omega }^6+108m{\omega }^6-16{\omega }^6).\\ & u\left(x,t\right)=B+C{\mathrm{cn}}^2\left(\sqrt{\omega }x-\mathit{\lambda t}\right)+{\xi }_0\mid m).\end{array}$$E12

$$\begin{array}{l}B=-\frac{4}{3}\left(2m-1\right){\omega }^2,C=4m{\omega }^2.\\ \lambda =\frac{128}{3}\left(m-2\right)\left(m+1\right)\left(2m-1\right){\omega }^7.\\ u\left(x,t\right)=B+C{\mathrm{cn}}^2\left(\sqrt{\omega }x-\mathit{\lambda t}\right)+{\xi }_0\mid m).\end{array}$$E13

• Lax Eq. (3):

$$\begin{array}{ll}C=& 2m{\omega }^2.\\ \lambda =& 4\omega (35B^3+140B^{2}m{\omega }^2-70B^2{\omega }^2+196{\mathit{Bm}}^2{\omega }^4-196\mathit{Bm}{\omega }^4+56B{\omega }^4+96m^3{\omega }^6\\ & -144m^2{\omega }^6+80m{\omega }^6-16{\omega }^6)u\left(x,t\right)=B+2\mathit{m\omega }{\mathrm{cn}}^2\left(\sqrt{\omega }\left(x-\mathit{\lambda t}\right)+{\xi }_0|m\right).\end{array}$$E14

• Kaup-Kuperschmidt Eq. (4):

$$\begin{array}{l}B=-\frac{1}{6}\left(2m-1\right){\omega }^2,C=\frac{m{\omega }^2}{2}.\\ \lambda =\frac{2}{3}\left(m-2\right)\left(m+1\right)\left(2m-1\right){\omega }^7.\\ u\left(x,t\right)=B+2\mathit{m\omega }{\mathrm{cn}}^2\left(\sqrt{\omega }\left(x-\mathit{\lambda t}\right)+{\xi }_0|m\right).\end{array}$$E15

• Letting m = 1 in Eqs. (2)–(4), we obtain solitonic solutions.

3. Soliton solutions

3.1 First family

Let

$$\begin{array}{l}A=\frac{252}{\alpha +\beta +\gamma },w=k^7.\\ c=-\frac{1}{126}\left(\alpha +\beta +\gamma \right)\left(\alpha -5\left(\beta +4\gamma \right)\right)-b-d.\\ d=\frac{42a}{\alpha +\beta +\gamma }+\frac{1}{378}\left(\alpha +\beta +\gamma \right)\left(\alpha -5\beta +10\gamma \right)+\frac{b}{3}.\end{array}$$E16

We make the transformation

$$u=A{\partial }_{x,x}log\left(1+exp\left(\mathit{kx}-\mathit{wt}\right)\right)$$E17

to obtain the soliton solution

$$u_{\text{soliton}}\left(x,t\right)=\frac{126k^2}{\left(\alpha +\beta +\gamma \right)\left(1+cosh\left(\mathit{kx}-k^{7}t-\xi \right)\right)}.$$E18

For a graphical illustration, see Figure 2.

Figure 2.

Soliton solution $u\left(x,t\right)=\frac{0.04}{1+cosh\left(0.0000128t-0.2x\right)}$ to the Sawada-Kotera-Ito equation with k = 0.2.

We are interested in the existence of two soliton solutions. Let

$$\begin{array}{l}u\left(x,t\right)=\frac{252}{\alpha +\beta +\gamma }{\partial }_{x,x}log\left(1+exp{\theta }_1+exp{\theta }_2+\rho exp\left({\theta }_1+{\theta }_2\right)\right).\\ {\theta }_1=k_{1}x-k_1^{7}t\text{and}{\theta }_2=k_{2}x-k_2^{7}t\end{array}$$E19

Making the choices

$$\begin{array}{l}a=-\frac{{\gamma }^2\left(\beta +2\gamma \right)\left({\beta }^3-3{\beta }^2\gamma -9{\mathit{\beta \gamma }}^2+31{\gamma }^3\right)}{49{\left(\beta -5\gamma \right)}^3}\\ b=-\frac{\left(\beta -\gamma \right)\left({\beta }^3-6{\beta }^2\gamma +15{\mathit{\beta \gamma }}^2-23{\gamma }^3\right)}{7{\left(\beta -5\gamma \right)}^2}\\ c=\frac{{\beta }^4-9{\beta }^3\gamma +25{\beta }^2{\gamma }^2-33{\mathit{\beta \gamma }}^3+76{\gamma }^4}{7{\left(\beta -5\gamma \right)}^2}\\ d=\frac{2\gamma \left({\beta }^3-5{\beta }^2\gamma +2{\mathit{\beta \gamma }}^2+17{\gamma }^3\right)}{7{\left(\beta -5\gamma \right)}^2}\\ \alpha =-\frac{{\beta }^2-4\mathit{\beta \gamma }+13{\gamma }^2}{\beta -5\gamma }\\ \rho =\frac{{\gamma }^2{\left(k_1-k_2\right)}^2{\left({k_1}^2-k_1{k}_2+{k_2}^2\right)}^2}{{\left(k_1+{\mathit{\text{extk}}}_2\right)}^2\left(2{\beta }^2{k_1}^2{k_2}^2-2\mathit{\beta \gamma }{k}_1{k}_2\left({k_1}^2+4k_1{k}_2+{k_2}^2\right)+{\gamma }^2\left({k_1}^4+4{k_1}^3{k}_2+9{k_1}^2{k_2}^2+4k_1{k_2}^3+{k_2}^4\right)\right)}.\end{array}$$E20

We obtain

$$\begin{array}{l}{u}_t+{\mathit{au}}^3{u}_x+{\mathit{bu}}_x^3+{\mathit{cuu}}_x{u}_{2x}+{\mathit{du}}^2{u}_{3x}+\alpha u_{2x}{u}_{3x}+\beta u_x{u}_{4x}+\gamma u_{5x}+u_{7x}=\\ \left(\beta -\gamma \right)\left(\beta -2\gamma \right)\left(\beta -3\gamma \right)R\left(k_1,k_2,{\theta }_1,{\theta }_2\right).\end{array}$$

We conclude that the two soliton solutions exist for the parameter values in Eq. (20) under the condition

$$\left(\beta -\gamma \right)\left(\beta -2\gamma \right)\left(\beta -3\gamma \right)=0.$$E21

We obtained the following result:

Theorem. The following families of KdV7 admit one and two soliton solutions for any p. The two soliton solutions have the form

$$u_{2-\text{soliton}}\left(x,t\right)=\frac{252}{\alpha +\beta +\gamma }{\partial }_{\mathit{xx}}log\left(1+exp\left({\theta }_1\right)+exp\left({\theta }_2\right)+\rho exp\left({\theta }_1+{\theta }_2\right)\right),$$E22

being

$${\theta }_1=k_{1}x-k_1^{7}t,{\theta }_2=k_{2}x-k_2^{7}t.$$E23

• First set:

$$\left\{\begin{array}{c}a=\frac{15p^3}{784},b=0,c=\frac{15p^2}{28},d=\frac{15p^2}{56},\alpha =\frac{5p}{2},\beta =p,\\ \rho =\frac{{\left(k_1-k_2\right)}^2{\left({k_1}^2-k_1{k}_2+{k_2}^2\right)}^2}{{\left(k_1+k_2\right)}^2{\left({k_1}^2+k_1{k}_2+{k_2}^2\right)}^2},\gamma =p\end{array}\right\}$$E24

KdV7:

$$u_t+\frac{15}{784}{p}^3{u}^3{u}_x+\frac{15}{28}{p}^2{\mathit{uu}}_x{u}_{2x}+\frac{15}{56}{p}^2{u}^2{u}_{3x}+\frac{5}{2}{\mathit{pu}}_{2x}{u}_{3x}+{\mathit{pu}}_x{u}_{4x}+{\mathit{puu}}_{5x}+u_{7x}=0$$E25

An illustration with p = 1 is shown in Figure 3. The solution is

Figure 3.

Two soliton solution for p = 1.

$$u\left(x,t\right)=\frac{14.e^{0.352855t+0.5x}+27.44e^{0.278313t+0.7x}+2.52676e^{0.2705t+1.2x}+0.0975793e^{0.262688t+1.7x}+0.0497854e^{0.188146t+1.9x}}{{\left(e^{0.172521t+0.5x}+e^{0.0979793t+0.7x}+0.0035561e^{0.0901668t+1.2x}+e^{0.180334t}\right)}^2}$$E26

• Second set:

$$\left\{\begin{array}{c}a=\frac{4p^3}{147},b=\frac{p^2}{7},c=\frac{6p^2}{7},d=\frac{2p^2}{7},\alpha =3p,\beta =2p,\\ \rho =\frac{{\left(k_1-k_2\right)}^2\left({k_1}^2-k_1{k}_2+{k_2}^2\right)}{{\left(k_1+k_2\right)}^2\left({k_1}^2+k_1{k}_2+{k_2}^2\right)},\gamma =p\end{array}\right\}$$E27

KdV7:

$$u_t+\frac{4}{147}{p}^3{u}^3{u}_x+\frac{1}{7}{p}^2{u}_x^3+\frac{6}{7}{p}^2{\mathit{uu}}_x{u}_{2x}+\frac{2}{7}{p}^2{u}^2{u}_{3x}+3{\mathit{pu}}_{2x}{u}_{3x}+2{\mathit{pu}}_x{u}_{4x}+{\mathit{puu}}_{5x}+u_{7x}=0$$E28

• Third set:

$$\left\{\begin{array}{c}a=\frac{5p^3}{98},b=\frac{5p^2}{14},c=\frac{10p^2}{7},d=\frac{5p^2}{14},\alpha =5p,\beta =3p,\\ \rho =\frac{{\left(k_1-k_2\right)}^2}{{\left(k_1+k_2\right)}^2},\gamma =p\end{array}\right\}$$E29

KdV7:

$$u_t+\frac{5}{98}{p}^3{u}^3{u}_x+\frac{5}{14}{p}^2{u}_x^3+\frac{10}{7}{p}^2{\mathit{uu}}_x{u}_{2x}+\frac{5}{14}{p}^2{u}^2{u}_{3x}+5{\mathit{pu}}_{2x}{u}_{3x}+3{\mathit{pu}}_x{u}_{4x}+{\mathit{puu}}_{5x}+u_{7x}=0$$E30

Now, our aim is to find three soliton solutions for the parameter values in Eq. (20). Assume the ansatz

$$u\left(x,t\right)=\frac{252}{\alpha +\beta +\gamma }{\partial }_{\mathit{xx}}log\left(1+\sum _{j=1}^3{\eta }_j+\sum _{1\le i<j\le 3}{\rho }_{i,j}{\eta }_i{\eta }_j+{\rho }_{\mathrm{1,2,3}}{\eta }_1{\eta }_2{\eta }_3\right),$$E31

where

$${\eta }_j=exp\left(k_{j}x-k_j^{7}t\right)\text{for}j=\mathrm{1,2,3}.$$E32

We have:

$$\begin{array}{l}{u}_t+{\mathit{au}}^3{u}_x+{\mathit{bu}}_x^3+{\mathit{cuu}}_x{u}_{2x}+{\mathit{du}}^2{u}_{3x}+\alpha u_{2x}{u}_{3x}+\beta u_x{u}_{4x}+\gamma {\mathit{uu}}_{5x}+u_{7x}=\\ 98\left(\beta -5\gamma \right)k_1{k}_2\left(k_1+k_2\right)({\gamma }^2{k}_1^6-{\gamma }^2{\rho }_{1,2}{k}_1^6-6{\gamma }^2{k}_2{k}_1^5+2\mathit{\beta \gamma }{k}_2{k}_1^5-4{\gamma }^2{k}_2{\rho }_{1,2}{k}_1^5\\ +2{\beta }^2{k}_2^2{k}_1^4+18{\gamma }^2{k}_2^2{k}_1^4-12\mathit{\beta \gamma }{k}_2^2{k}_1^4-8{\gamma }^2{k}_2^2{\rho }_{1,2}{k}_1^4-4{\beta }^2{k}_2^3{k}_1^3-26{\gamma }^2{k}_2^3{k}_1^3+20\mathit{\beta \gamma }{k}_2^3{k}_1^3\\ -10{\gamma }^2{k}_2^3{\rho }_{1,2}{k}_1^3+2{\beta }^2{k}_2^4{k}_1^2+18{\gamma }^2{k}_2^4{k}_1^2-12\mathit{\beta \gamma }{k}_2^4{k}_1^2-8{\gamma }^2{k}_2^4{\rho }_{1,2}{k}_1^2-6{\gamma }^2{k}_2^5{k}_1+2\mathit{\beta \gamma }{k}_2^5{k}_1\\ -4{\gamma }^2{k}_2^5{\rho }_{1,2}{k}_1+{\gamma }^2{k}_2^6-{\gamma }^2{k}_2^6{\rho }_{1,2})/{\gamma }^4{z}_1{z}_2\\ +98\left(\beta -5\gamma \right)k_1{k}_3\left(k_1+k_3\right)({\gamma }^2{k}_1^6-{\gamma }^2{\rho }_{1,3}{k}_1^6-6{\gamma }^2{k}_3{k}_1^5+2\mathit{\beta \gamma }{k}_3{k}_1^5-4{\gamma }^2{k}_3{\rho }_{1,3}{k}_1^5\\ +2{\beta }^2{k}_3^2{k}_1^4+18{\gamma }^2{k}_3^2{k}_1^4-12\mathit{\beta \gamma }{k}_3^2{k}_1^4-8{\gamma }^2{k}_3^2{\rho }_{1,3}{k}_1^4-4{\beta }^2{k}_3^3{k}_1^3-26{\gamma }^2{k}_3^3{k}_1^3+20\mathit{\beta \gamma }{k}_3^3{k}_1^3-\\ 10{\gamma }^2{k}_3^3{\rho }_{1,3}{k}_1^3+2{\beta }^2{k}_3^4{k}_1^2+18{\gamma }^2{k}_3^4{k}_1^2-12\mathit{\beta \gamma }{k}_3^4{k}_1^2-8{\gamma }^2{k}_3^4{\rho }_{1,3}{k}_1^2-6{\gamma }^2{k}_3^5{k}_1+2\mathit{\beta \gamma }{k}_3^5{k}_1\\ -4{\gamma }^2{k}_3^5{\rho }_{1,3}{k}_1+{\gamma }^2{k}_3^6-{\gamma }^2{k}_3^6{\rho }_{1,3})/{\gamma }^4{z}_1{z}_3\\ +98\left(\beta -5\gamma \right)k_2{k}_3\left(k_2+k_3\right)({\gamma }^2{k}_2^6-{\gamma }^2{\rho }_{2,3}{k}_2^6-6{\gamma }^2{k}_3{k}_2^5+2\mathit{\beta \gamma }{k}_3{k}_2^5-4{\gamma }^2{k}_3{\rho }_{2,3}{k}_2^5\\ +2{\beta }^2{k}_3^2{k}_2^4+18{\gamma }^2{k}_3^2{k}_2^4-12\mathit{\beta \gamma }{k}_3^2{k}_2^4-8{\gamma }^2{k}_3^2{\rho }_{2,3}{k}_2^4-4{\beta }^2{k}_3^3{k}_2^3-26{\gamma }^2{k}_3^3{k}_2^3+20\mathit{\beta \gamma }{k}_3^3{k}_2^3\\ -10{\gamma }^2{k}_3^3{\rho }_{2,3}{k}_2^3+2{\beta }^2{k}_3^4{k}_2^2+18{\gamma }^2{k}_3^4{k}_2^2-12\mathit{\beta \gamma }{k}_3^4{k}_2^2-8{\gamma }^2{k}_3^4{\rho }_{2,3}{k}_2^2-6{\gamma }^2{k}_3^5{k}_2+2\mathit{\beta \gamma }{k}_3^5{k}_2\\ -4{\gamma }^2{k}_3^5{\rho }_{2,3}{k}_2+{\gamma }^2{k}_3^6-{\gamma }^2{k}_3^6{\rho }_{2,3})/{\gamma }^4{z}_2{z}_3+\mathrm{h}.\mathrm{o}.\mathrm{t}\end{array}$$

Equating to zero the coefficients of $z_1{z}_2$, $z_1{z}_3$_, and $z_2{z}_3$ and solving the resulting system of algebraic equations we obtain

$$\begin{array}{l}{\rho }_{1,2}=\frac{{\left(k_1-k_2\right)}^2\left(2{\beta }^2{k}_2^2{k}_1^2+2\mathit{\beta \gamma }{k}_2{k}_1^3-8\mathit{\beta \gamma }{k}_2^2{k}_1^2+2\mathit{\beta \gamma }{k}_2^3{k}_1+{\gamma }^2{k}_1^4-4{\gamma }^2{k}_2{k}_1^3+9{\gamma }^2{k}_2^2{k}_1^2-4{\gamma }^2{k}_2^3{k}_1+{\gamma }^2{k}_2^4\right)}{{\gamma }^2{\left(k_1+k_2\right)}^2{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2}.\\ {\rho }_{1,3}=\frac{{\left(k_1-k_3\right)}^2\left(2{\beta }^2{k}_3^2{k}_1^2+2\mathit{\beta \gamma }{k}_3{k}_1^3-8\mathit{\beta \gamma }{k}_3^2{k}_1^2+2\mathit{\beta \gamma }{k}_3^3{k}_1+{\gamma }^2{k}_1^4-4{\gamma }^2{k}_3{k}_1^3+9{\gamma }^2{k}_3^2{k}_1^2-4{\gamma }^2{k}_3^3{k}_1+{\gamma }^2{k}_3^4\right)}{{\gamma }^2{\left(k_1+k_3\right)}^2{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2}.\\ {\rho }_{2,3}=\frac{{\left(k_2-k_3\right)}^2\left(2{\beta }^2{k}_3^2{k}_2^2+2\mathit{\beta \gamma }{k}_3{k}_2^3-8\mathit{\beta \gamma }{k}_3^2{k}_2^2+2\mathit{\beta \gamma }{k}_3^3{k}_2+{\gamma }^2{k}_2^4-4{\gamma }^2{k}_3{k}_2^3+9{\gamma }^2{k}_3^2{k}_2^2-4{\gamma }^2{k}_3^3{k}_2+{\gamma }^2{k}_3^4\right)}{{\gamma }^2{\left(k_2+k_3\right)}^2{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2}.\end{array}$$E33

Next, we equate to zero the coefficient of $z_1{z}_2{z}_3$ in order to obtain the value for ${\rho }_{\mathrm{1,2,3}}$. It is given as follows.

• For $\beta =\gamma$:

$$\begin{array}{l}{\rho }_{\mathrm{1,2,3}}=P_1/Q_1,\text{where}\\ P_1={\gamma }^4{\left(k_2-k_1\right)}^2{\left(k_2-k_3\right)}^2{\left(k_3-k_1\right)}^2\left(k_1+k_2+k_3\right)\\ (k_2^4{k}_1^{13}+k_3^4{k}_1^{13}-2k_2{k}_3^3{k}_1^{13}+3k_2^2{k}_3^2{k}_1^{13}-2k_2^3{k}_3{k}_1^{13}+k_2^5{k}_1^{12}+k_3^5{k}_1^{12}-k_2{k}_3^4{k}_1^{12}+k_2^2{k}_3^3{k}_1^{12}\\ +k_2^3{k}_3^2{k}_1^{12}-k_2^4{k}_3{k}_1^{12}+2k_2^6{k}_1^{11}+2k_3^6{k}_1^{11}-3k_2{k}_3^5{k}_1^{11}+6k_2^2{k}_3^4{k}_1^{11}-5k_2^3{k}_3^3{k}_1^{11}+6k_2^4{k}_3^2{k}_1^{11}\\ -3k_2^5{k}_3{k}_1^{11}+2k_2^7{k}_1^{10}+2k_3^7{k}_1^{10}-4k_2{k}_3^6{k}_1^{10}-10k_2^2{k}_3^5{k}_1^{10}-18k_2^3{k}_3^4{k}_1^{10}-18k_2^4{k}_3^3{k}_1^{10}-10k_2^5{k}_3^2{k}_1^{10}\\ -4k_2^6{k}_3{k}_1^{10}+3k_2^8{k}_1^9+3k_3^8{k}_1^9-6k_2{k}_3^7{k}_1^9-7k_2^2{k}_3^6{k}_1^9-38k_2^3{k}_3^5{k}_1^9-26k_2^4{k}_3^4{k}_1^9-38k_2^5{k}_3^3{k}_1^9-7k_2^6{k}_3^2{k}_1^9\\ -6k_2^7{k}_3{k}_1^9+3k_2^9{k}_1^8+3k_3^9{k}_1^8-3k_2{k}_3^8{k}_1^8-7k_2^2{k}_3^7{k}_1^8-35k_2^3{k}_3^6{k}_1^8-58k_2^4{k}_3^5{k}_1^8-58k_2^5{k}_3^4{k}_1^8-35k_2^6{k}_3^3{k}_1^8\\ -7k_2^7{k}_3^2{k}_1^8-3k_2^8{k}_3{k}_1^8+2k_2^{10}{k}_1^7+2k_3^{10}{k}_1^7-6k_2{k}_3^9{k}_1^7-7k_2^2{k}_3^8{k}_1^7-41k_2^3{k}_3^7{k}_1^7-60k_2^4{k}_3^6{k}_1^7-99k_2^5{k}_3^5{k}_1^7\\ -60k_2^6{k}_3^4{k}_1^7-41k_2^7{k}_3^3{k}_1^7-7k_2^8{k}_3^2{k}_1^7-6k_2^9{k}_3{k}_1^7+2k_2^{11}{k}_1^6+2k_3^{11}{k}_1^6-4k_2{k}_3^{10}{k}_1^6-7k_2^2{k}_3^9{k}_1^6-35k_2^3{k}_3^8{k}_1^6\\ -60k_2^4{k}_3^7{k}_1^6-92k_2^5{k}_3^6{k}_1^6-92k_2^6{k}_3^5{k}_1^6-60k_2^7{k}_3^4{k}_1^6-35k_2^8{k}_3^3{k}_1^6-7k_2^9{k}_3^2{k}_1^6-4k_2^{10}{k}_3{k}_1^6+k_2^{12}{k}_1^5+k_3^{12}{k}_1^5\\ -3k_2{k}_3^{11}{k}_1^5-10k_2^2{k}_3^{10}{k}_1^5-38k_2^3{k}_3^9{k}_1^5-58k_2^4{k}_3^8{k}_1^5-99k_2^5{k}_3^7{k}_1^5-92k_2^6{k}_3^6{k}_1^5-99k_2^7{k}_3^5{k}_1^5-58k_2^8{k}_3^4{k}_1^5\\ -38k_2^9{k}_3^3{k}_1^5-10k_2^{10}{k}_3^2{k}_1^5-3k_2^{11}{k}_3{k}_1^5+k_2^{13}{k}_1^4+k_3^{13}{k}_1^4-k_2{k}_3^{12}{k}_1^4+6k_2^2{k}_3^{11}{k}_1^4-18k_2^3{k}_3^{10}{k}_1^4-26k_2^4{k}_3^9{k}_1^4\\ -58k_2^5{k}_3^8{k}_1^4-60k_2^6{k}_3^7{k}_1^4-60k_2^7{k}_3^6{k}_1^4-58k_2^8{k}_3^5{k}_1^4-26k_2^9{k}_3^4{k}_1^4-18k_2^{10}{k}_3^3{k}_1^4+6k_2^{11}{k}_3^2{k}_1^4-k_2^{12}{k}_3{k}_1^4\\ -2k_2{k}_3^{13}{k}_1^3+k_2^2{k}_3^{12}{k}_1^3-5k_2^3{k}_3^{11}{k}_1^3-18k_2^4{k}_3^{10}{k}_1^3-38k_2^5{k}_3^9{k}_1^3-35k_2^6{k}_3^8{k}_1^3-41k_2^7{k}_3^7{k}_1^3-35k_2^8{k}_3^6{k}_1^3\\ -38k_2^9{k}_3^5{k}_1^3-18k_2^{10}{k}_3^4{k}_1^3-5k_2^{11}{k}_3^3{k}_1^3+k_2^{12}{k}_3^2{k}_1^3-2k_2^{13}{k}_3{k}_1^3+3k_2^2{k}_3^{13}{k}_1^2+k_2^3{k}_3^{12}{k}_1^2+6k_2^4{k}_3^{11}{k}_1^2\\ -10k_2^5{k}_3^{10}{k}_1^2-7k_2^6{k}_3^9{k}_1^2-7k_2^7{k}_3^8{k}_1^2-7k_2^8{k}_3^7{k}_1^2-7k_2^9{k}_3^6{k}_1^2-10k_2^{10}{k}_3^5{k}_1^2+6k_2^{11}{k}_3^4{k}_1^2+k_2^{12}{k}_3^3{k}_1^2\\ +3k_2^{13}{k}_3^2{k}_1^2-2k_2^3{k}_3^{13}{k}_1-k_2^4{k}_3^{12}{k}_1-3k_2^5{k}_3^{11}{k}_1-4k_2^6{k}_3^{10}{k}_1-6k_2^7{k}_3^9{k}_1-3k_2^8{k}_3^8{k}_1-6k_2^9{k}_3^7{k}_1-4k_2^{10}{k}_3^6{k}_1\\ -3k_2^{11}{k}_3^5{k}_1-k_2^{12}{k}_3^4{k}_1-2k_2^{13}{k}_3^3{k}_1+k_2^4{k}_3^{13}+k_2^5{k}_3^{12}+2k_2^6{k}_3^{11}+2k_2^7{k}_3^{10}+3k_2^8{k}_3^9+3k_2^9{k}_3^8+2k_2^{10}{k}_3^7\\ +2k_2^{11}{k}_3^6+k_2^{12}{k}_3^5+k_2^{13}{k}_3^4),\end{array}$$

and

$$\begin{array}{l}{Q}_1={\gamma }^4{\left(k_1+k_2\right)}^2{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2{\left(k_1+k_2+k_3\right)}^2{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2\\ {\left(k_2^2+k_3{k}_2+k_3^2\right)}^2\\ (k_1^4+2k_2{k}_1^3+2k_3{k}_1^3+3k_2^2{k}_1^2+3k_3^2{k}_1^2+5k_2{k}_3{k}_1^2+2k_2^3{k}_1+2k_3^3{k}_1+5k_2{k}_3^2{k}_1+5k_2^2{k}_3{k}_1\\ +k_2^4+k_3^4+2k_2{k}_3^3+3k_2^2{k}_3^2+2k_2^3{k}_3).\end{array}$$

• For $\beta =2\gamma$:

$$\begin{array}{l}{\rho }_{\mathrm{1,2,3}}=P_2/Q_2,\text{where}\\ P_2={\gamma }^4{\left(k_2-k_1\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right)\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_2-k_3\right)}^2{\left(k_3-k_1\right)}^2{\left(k_1+k_2+k_3\right)}^2\\ \left(k_1^2-k_3{k}_1+k_3^2\right)\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2-k_3{k}_2+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)(k_1^4+2k_2{k}_1^3+2k_3{k}_1^3\\ +3k_2^2{k}_1^2+3k_3^2{k}_1^2+5k_2{k}_3{k}_1^2+2k_2^3{k}_1+2k_3^3{k}_1+5k_2{k}_3^2{k}_1+5k_2^2{k}_3{k}_1+k_2^4+k_3^4+2k_2{k}_3^3+3k_2^2{k}_3^2+2k_2^3{k}_3),\end{array}$$

and

$$\begin{array}{l}{Q}_2={\gamma }^4{\left(k_1+k_2\right)}^2{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2{\left(k_1+k_2+k_3\right)}^2{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2\\ {\left(k_2^2+k_3{k}_2+k_3^2\right)}^2\\ (k_1^4+2k_2{k}_1^3+2k_3{k}_1^3+3k_2^2{k}_1^2+3k_3^2{k}_1^2+5k_2{k}_3{k}_1^2+2k_2^3{k}_1+2k_3^3{k}_1+5k_2{k}_3^2{k}_1+5k_2^2{k}_3{k}_1+k_2^4+k_3^4\\ +2k_2{k}_3^3+3k_2^2{k}_3^2+2k_2^3{k}_3).\end{array}$$

• For $\beta =3\gamma$:

$$\begin{array}{l}{\rho }_{\mathrm{1,2,3}}=P_3/Q_3,\text{where}\\ P_3={\gamma }^4{\left(k_2-k_1\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right)\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_2-k_3\right)}^2{\left(k_3-k_1\right)}^2{\left(k_1+k_2+k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\\ \left(k_1^2+k_3{k}_1+k_3^2\right)\\ \left(k_2^2-k_3{k}_2+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)(k_1^4+2k_2{k}_1^3+2k_3{k}_1^3+3k_2^2{k}_1^2+3k_3^2{k}_1^2+5k_2{k}_3{k}_1^2+2k_2^3{k}_1+2k_3^3{k}_1+5k_2{k}_3^2{k}_1\\ +5k_2^2{k}_3{k}_1+k_2^4+k_3^4+2k_2{k}_3^3+3k_2^2{k}_3^2+2k_2^3{k}_3),\end{array}$$

and

$$\begin{array}{l}{Q}_3={\gamma }^4{\left(k_1+k_2\right)}^2{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2{\left(k_1+k_2+k_3\right)}^2{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2\\ (k_1^4+2k_2{k}_1^3+2k_3{k}_1^3+3k_2^2{k}_1^2+3k_3^2{k}_1^2+5k_2{k}_3{k}_1^2+2k_2^3{k}_1+2k_3^3{k}_1+5k_2{k}_3^2{k}_1+5k_2^2{k}_3{k}_1+k_2^4+k_3^4\\ +2k_2{k}_3^3+3k_2^2{k}_3^2+2k_2^3{k}_3).\end{array}$$

We have three soliton solutions only when $\beta =2\gamma$ or $\beta =3\gamma$. Thus, the KdV7 has two soliton solutions for the parameter values in Eq. (20), but it does not have three soliton solutions for $\gamma =\beta .$.

3.2 Second family

Let

$$\begin{array}{l}a=-\frac{5\left(7\alpha +5\beta -6\gamma \right){\left(\alpha +\beta +\gamma \right)}^2}{7938}.\\ b=\frac{1}{63}\left(\alpha +\beta +\gamma \right)\left(36\alpha +35\beta +10\gamma \right).\\ c=-\frac{1}{63}\left(\alpha +\beta +\gamma \right)\left(37\alpha +35\beta +15\gamma \right).\\ d=\frac{1}{126}\left(\alpha +\beta +\gamma \right)\left(\alpha +5\beta +30\gamma \right).\end{array}$$E34

We make the transformation

$$u\left(x,t\right)=\frac{126}{\alpha +\beta +\gamma }{\partial }_{x,x}log\left(1+exp\left(\mathit{kx}-k^{7}t\right)+\rho exp\left(2\mathit{kx}-2k^{7}t\right)\right)$$E35

to obtain the soliton solution

$$u_{\text{soliton}}\left(x,t\right)=\frac{504k^2{e}^{k^{7}t+\mathit{kx}}}{\left(\alpha +\beta +\gamma \right){\left(2e^{k^{7}t}+e^{\mathit{kx}}\right)}^2}\text{for}\rho =\frac{1}{4}.$$E36

A cnoidal wave solution is

$$u_{\text{cnoidal}}\left(x,t\right)=p+\frac{\left(1-2m\pm \sqrt{m^2-m+1}\right)p}{m-1}{\mathrm{cn}}^2\left(\sqrt{\frac{q\left(\alpha +\beta +\gamma \right)}{252m}}\left(x-\mathit{\lambda t}\right)+{\xi }_0,m\right),$$E37

where

$$\begin{array}{l}\lambda =-\frac{1}{500094m^3}\left[{\left(\alpha +\beta +\gamma \right)}^2\right(2205\alpha m^3{p}^3+1575\beta m^3{p}^3-1890\gamma m^3{p}^3-126\alpha m^3{p}^{2}q\\ -630\beta m^3{p}^{2}q-3780\gamma m^3{p}^{2}q-2331\alpha m^3{\mathit{pq}}^2-2205\beta m^3{\mathit{pq}}^2-2016\gamma m^3{\mathit{pq}}^2+2\alpha m^3{q}^3\\ +2\beta m^3{q}^3-124\gamma m^3{q}^3+63\alpha m^2{p}^{2}q+315\beta m^2{p}^{2}q+1890\gamma m^2{p}^{2}q+2331\alpha m^2{\mathit{pq}}^2+2205\beta m^2{\mathit{pq}}^2\\ +2016\gamma m^2{\mathit{pq}}^2-3\alpha m^2{q}^3-3\beta m^2{q}^3+186\gamma m^2{q}^3-126\gamma {\mathit{mpq}}^2-3\alpha {\mathit{mq}}^3-3\beta {\mathit{mq}}^3\\ -66\gamma {\mathit{mq}}^3+2\alpha q^3+2\beta q^3+2\gamma q^3\left)\right].\end{array}$$

3.3 Third family

Let

$$\begin{array}{c}a=\frac{8{\left(\alpha +\beta +\gamma \right)}^2\left(4\alpha -10\beta +25\gamma \right)}{453789}.\\ b=\frac{1}{882}\left(-16{\alpha }^2+22\mathit{\alpha \beta }-23\mathit{\alpha \gamma }+38{\beta }^2+31\mathit{\beta \gamma }-7{\gamma }^2\right).\\ c=\frac{8{\alpha }^2+2\mathit{\alpha \beta }+107\mathit{\alpha \gamma }-6{\beta }^2+93\mathit{\beta \gamma }+99{\gamma }^2}{1029}.\\ d=-\frac{2\left(2{\alpha }^2+4\mathit{\alpha \beta }-31\mathit{\alpha \gamma }+2{\beta }^2-31\mathit{\beta \gamma }-33{\gamma }^2\right)}{1029}.\end{array}$$E38

We make the transformation

$$u\left(x,t\right)=\frac{441}{2\left(\alpha +\beta +\gamma \right)}{\partial }_{x,x}log\left(1+exp\left(\mathit{kx}-k^{7}t\right)+\rho exp\left(2\mathit{kx}-2k^{7}t\right)\right)$$E39

to obtain the soliton solution

$$u_{\text{soliton}}\left(x,t\right)=\frac{3528k^2\left(16e^{k^{7}t-\mathit{kx}}+e^{\mathit{kx}-k^{7}t}+4\right)}{\left(\alpha +\beta +\gamma \right){\left(16e^{k^{7}t-\mathit{kx}}+e^{\mathit{kx}-k^{7}t}+16\right)}^2}\text{for}\rho =\frac{1}{16}.$$E40

A cnoidal wave solution is

$$u\left(x,t\right)=p+\frac{3\mathit{mp}}{1-2m}{\mathrm{cn}}^2\left(\sqrt{\frac{2p\left(\alpha +\beta +\gamma \right)}{147\left(1-2m\right)}}\left(x-\mathit{\lambda t}\right)+{\xi }_0,m\right),$$E41

where

$$\lambda =-\frac{4\left(m-2\right)\left(m+1\right)p^3\left(48\alpha -50\beta -99\gamma \right){\left(\alpha +\beta +\gamma \right)}^2}{3176523{\left(2m-1\right)}^2}.$$E42

Let us investigate the existence of two soliton solutions in the ansatz form

$$\begin{array}{l}u\left(x,t\right)=\frac{441}{2\left(\alpha +\beta +\gamma \right)}{\partial }_{\mathit{xx}}log(1+exp{\theta }_1+exp{\theta }_2+\frac{1}{16}\left[exp\left(2{\theta }_1\right)+exp\left(2{\theta }_2\right)\right]+\\ \rho \left[exp\left({\theta }_1+2{\theta }_2\right)+exp\left(2{\theta }_1+{\theta }_2\right)\right]+\kappa exp\left({\theta }_1+{\theta }_2\right)+{\rho }^{2}exp\left(2{\theta }_1+2{\theta }_2\right)),\\ \text{where}{\theta }_1=k_{1}x-k_1^{7}t\text{and}{\theta }_2=k_{2}x-k_2^{7}t.\end{array}$$

We have three soliton solutions for the following choices:

$$a=\frac{4{\gamma }^3}{147},b=\frac{5{\gamma }^2}{14},c=\frac{9{\gamma }^2}{7},d=\frac{2{\gamma }^2}{7},\alpha =6\gamma ,\beta =\frac{7\gamma }{2}.$$E43

$$\begin{array}{cc}\kappa =\frac{2k_1^4-k_2^2{k}_1^2+2k_2^4}{2{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)},& \rho =\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right)}{16{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)}\end{array}$$E44

Letting $\gamma =42$ gives the Kaup–Kuperschmidt seventh-order equation:

$$u_t+2016u^3{u}_x+630u_x^3+2268u_x{u}_{2x}+504u^2{u}_{3x}+252u_{2x}{u}_{3x}+147u_x{u}_{4x}+42{\mathit{uu}}_{5x}+u_{7x}=0.$$E45

3.4 Three soliton solutions

The three soliton solutions have the form

$$u_{3-\text{soliton}}\left(x,t\right)=1/2{\partial }_{\mathit{xx}}log\left(1+\sum _{i,j,l=0}^2{\rho }_{i,j,l}exp\left(i\left(k_{1}x-k_1^{7}t\right)+j\left(k_{2}x-k_2^{7}t\right)+l\left(k_{3}x-k_3^{7}t\right)\right)\right),$$E46

where ${\rho }_{i,j,l}=1$ when i + j + l = 1. The parameter values are obtained as follows. First, we set

$$k_{1}x-k_1^{7}t=log\left(z_1\right),k_{2}x-k_2^{7}t=log\left(z_2\right)\text{and}{k}_{3}x-k_3^{7}t=log\left(z_3\right)$$E47

to get

$$\begin{array}{l}{u}_t+2016u^3{u}_x+630u_x^3+2268u_x{u}_{2x}+504u^2{u}_{3x}\\ +252u_{2x}{u}_{3x}+147u_x{u}_{4x}+42u_{5x}+u_{7x}=\mathrm{\Psi }\left(z_1,z_2,z_3\right).\end{array}$$

Next, we solve the equation

$$\frac{{\partial }^{i+j+l}}{\partial z_1^i\partial z_2^j{\partial }^l{z}_3}\mathrm{\Psi }\left(z_1,z_2,z_3\right)\left|{}_{z_1=z_2=z_3=0}\right.=0$$E48

for ${\rho }_{i,j,l}$. The parameter values are:

$${\rho }_{\mathrm{0,0,2}}=\frac{1}{16},{\rho }_{\mathrm{0,1,1}}=\frac{2k_2^4-k_3^2{k}_2^2+2k_3^4}{2{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)},{\rho }_{\mathrm{0,1,2}}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)}{16{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)}.$$

$${\rho }_{\mathrm{0,2,0}}=\frac{1}{16},{\rho }_{\mathrm{0,2,1}}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)}{16{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)},{\rho }_{\mathrm{1,0,1}}=\frac{2k_1^4-k_3^2{k}_1^2+2k_3^4}{2{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)}.$$

$${\rho }_{\mathrm{1,0,2}}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)}{16{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)},{\rho }_{\mathrm{1,1,0}}=\frac{2k_1^4-k_2^2{k}_1^2+2k_2^4}{2{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)},{\rho }_{\mathrm{1,2,0}}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right)}{16{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)}.$$

$${\rho }_{\mathrm{2,0,0}}=\frac{1}{16},{\rho }_{\mathrm{2,0,1}}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)}{16{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)},{\rho }_{\mathrm{2,1,0}}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right)}{16{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)}.$$

$${\rho }_{\mathrm{0,2,2}}=\frac{{\left(k_2-k_3\right)}^4{\left(k_2^2-k_3{k}_2+k_3^2\right)}^2}{256{\left(k_2+k_3\right)}^4{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2}.$$

$${\rho }_{\mathrm{1,1,1}}=\frac{1}{4{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}$$

$$\begin{array}{l}\times [4k_2^4{k}_1^8+4k_3^4{k}_1^8-2k_2^2{k}_3^2{k}_1^8-2k_2^6{k}_1^6-2k_3^6{k}_1^6-k_2^2{k}_3^4{k}_1^6-k_2^4{k}_3^2{k}_1^6+4k_2^8{k}_1^4+4k_3^8{k}_1^4\\ -k_2^2{k}_3^6{k}_1^4-6k_2^4{k}_3^4{k}_1^4-k_2^6{k}_3^2{k}_1^4-2k_2^2{k}_3^8{k}_1^2-k_2^4{k}_3^6{k}_1^2-k_2^6{k}_3^4{k}_1^2-2k_2^8{k}_3^2{k}_1^2+4k_2^4{k}_3^8-2k_2^6{k}_3^6+4k_2^8{k}_3^4].\end{array}$$

$${\rho }_{\mathrm{1,1,2}}=\frac{\left(2k_1^4-k_2^2{k}_1^2+2k_2^4\right){\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(k_2^2-k_3{k}_2+k_3^2\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}.$$

$${\rho }_{\mathrm{1,2,1}}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)\left(2k_1^4-k_3^2{k}_1^2+2k_3^4\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}.$$

$${\rho }_{\mathrm{1,2,2}}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^4\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_2^2-k_3{k}_2+k_3^2\right)}^2}{256{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^4\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_2^2+k_3{k}_2+k_3^2\right)}^2}.$$

$${\rho }_{\mathrm{2,0,2}}=\frac{{\left(k_1-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2}{256{\left(k_1+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2}.$$

$${\rho }_{\mathrm{2,1,1}}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(2k_2^4-k_3^2{k}_2^2+2k_3^4\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}.$$

$${\rho }_{\mathrm{2,1,2}}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^2{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)}{256{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^4{\left(k_2+k_3\right)}^2{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)}.$$

$${\rho }_{\mathrm{2,2,0}}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2}{256{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2}.$$

$${\rho }_{\mathrm{2,2,1}}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(k_2^2-k_3{k}_2+k_3^2\right)}{256{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}.$$

$${\rho }_{\mathrm{2,2,2}}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2{\left(k_2^2-k_3{k}_2+k_3^2\right)}^2}{4096{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^4{\left(k_2+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2}.$$

3.5 Four soliton solutions

The four soliton solutions have the form $u\left(x,t\right)=1/2{\partial }_{\mathit{xx}}logf\left(x,t\right)$, where

$$\left\{\begin{array}{l}f\left(x,t\right)={\sum }_{i_1=0}^2{\sum }_{i_2=0}^2{\sum }_{i_3=0}^2{\sum }_{i_4=0}^2{B}_{i_1{i}_2{i}_3{i}_4}{z}_1^{i_1}{z}_2^{i_2}{z}_3^{i_3}{z}_4^{i_4},\\ z_j=exp\left(k_{j}x-k_j^{7}x\right)\text{for}\mathrm{any}j\end{array}\right.$$E49

The 76 coefficients $B_{i_1{i}_2{i}_3{i}_4}$ are given by

$$B_{0002}=\frac{1}{16},B_{0011}=\frac{2k_3^4-k_4^2{k}_3^2+2k_4^4}{2{\left(k_3+k_4\right)}^2\left(k_3^2+k_4{k}_3+k_4^2\right)}.$$

$$B_{0012}=\frac{{\left(k_3-k_4\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)}{16{\left(k_3+k_4\right)}^2\left(k_3^2+k_4{k}_3+k_4^2\right)},B_{0020}=\frac{1}{16}.$$

$$\begin{array}{l}{B}_{0021}=\frac{{\left(k_3-k_4\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)}{16{\left(k_3+k_4\right)}^2\left(k_3^2+k_4{k}_3+k_4^2\right)}.\\ B_{0022}=\frac{{\left(k_3-k_4\right)}^4{\left(k_3^2-k_4{k}_3+k_4^2\right)}^2}{256{\left(k_3+k_4\right)}^4{\left(k_3^2+k_4{k}_3+k_4^2\right)}^2}.\\ B_{0101}=\frac{2k_2^4-k_4^2{k}_2^2+2k_4^4}{2{\left(k_2+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)}.\\ B_{0102}=\frac{{\left(k_2-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)}{16{\left(k_2+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)}.\\ B_{0110}=\frac{2k_2^4-k_3^2{k}_2^2+2k_3^4}{2{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)}.\end{array}$$

$$\begin{array}{l}{B}_{0111}=[4k_3^4{k}_2^8+4k_4^4{k}_2^8-2k_3^2{k}_4^2{k}_2^8-2k_3^6{k}_2^6-2k_4^6{k}_2^6-k_3^2{k}_4^4{k}_2^6\\ -k_3^4{k}_4^2{k}_2^6+4k_3^8{k}_2^4+4k_4^8{k}_2^4-k_3^2{k}_4^6{k}_2^4-6k_3^4{k}_4^4{k}_2^4\\ -k_3^6{k}_4^2{k}_2^4-2k_3^2{k}_4^8{k}_2^2-k_3^4{k}_4^6{k}_2^2-k_3^6{k}_4^4{k}_2^2-2k_3^8{k}_4^2{k}_2^2\\ +4k_3^4{k}_4^8-2k_3^6{k}_4^6+4k_3^8{k}_4^4]/[4{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_2+k_4\right)}^2\\ {\left(k_3+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\\ B_{0112}=\frac{\left(2k_2^4-k_3^2{k}_2^2+2k_3^4\right){\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)}{32{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}\\ B_{0120}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)}{16{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)}\\ B_{0121}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_3-k_4\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)\left(2k_2^4-k_4^2{k}_2^2+2k_4^4\right)}{32{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}\end{array}$$

$$\begin{array}{l}\begin{array}{l}{B}_{0122}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^4\left(k_2^2-k_4{k}_2+k_4^2\right){\left(k_3^2-k_4{k}_3+k_4^2\right)}^2}{256{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^4\left(k_2^2+k_4{k}_2+k_4^2\right){\left(k_3^2+k_4{k}_3+k_4^2\right)}^2}\\ \begin{array}{l}{B}_{0200}=\frac{1}{16},B_{0201}=\frac{{\left(k_2-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)}{16{\left(k_2+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)}\\ B_{0202}=\frac{{\left(k_2-k_4\right)}^4{\left(k_2^2-k_4{k}_2+k_4^2\right)}^2}{256{\left(k_2+k_4\right)}^4{\left(k_2^2+k_4{k}_2+k_4^2\right)}^2},B_{0210}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)}{16{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)}.\\ \begin{array}{l}{B}_{0211}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_2-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)\left(2k_3^4-k_4^2{k}_3^2+2k_4^4\right)}{32{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}.\\ \begin{array}{l}{B}_{0212}=\frac{{\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_2-k_4\right)}^4{\left(k_3-k_4\right)}^2{\left(k_2^2-k_4{k}_2+k_4^2\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)}{256{\left(k_2+k_3\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_2+k_4\right)}^4{\left(k_3+k_4\right)}^2{\left(k_2^2+k_4{k}_2+k_4^2\right)}^2\left(k_3^2+k_4{k}_3+k_4^2\right)}.\\ B_{0220}=\frac{{\left(k_2-k_3\right)}^4{\left(k_2^2-k_3{k}_2+k_3^2\right)}^2}{256{\left(k_2+k_3\right)}^4{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2}-\\ B_{0221}=\frac{{\left(k_2-k_3\right)}^4{\left(k_2^2-k_3{k}_2+k_3^2\right)}^2{\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)}{256{\left(k_2+k_3\right)}^4{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}.\\ B_{0222}=\frac{{\left(k_2-k_3\right)}^4{\left(k_2^2-k_3{k}_2+k_3^2\right)}^2{\left(k_2-k_4\right)}^4{\left(k_3-k_4\right)}^4{\left(k_2^2-k_4{k}_2+k_4^2\right)}^2{\left(k_3^2-k_4{k}_3+k_4^2\right)}^2}{4096{\left(k_2+k_3\right)}^4{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2{\left(k_2+k_4\right)}^4{\left(k_3+k_4\right)}^4{\left(k_2^2+k_4{k}_2+k_4^2\right)}^2{\left(k_3^2+k_4{k}_3+k_4^2\right)}^2}.\\ B_{1001}=\frac{2k_1^4-k_4^2{k}_1^2+2k_4^4}{2{\left(k_1+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)},B_{1002}=\frac{{\left(k_1-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)}{16{\left(k_1+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)}.\\ B_{1010}=\frac{2k_1^4-k_3^2{k}_1^2+2k_3^4}{2{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)}\end{array}\end{array}\end{array}\end{array}\end{array}$$

$$\begin{array}{l}{B}_{1011}=[4k_3^4{k}_1^8+4k_4^4{k}_1^8-2k_3^2{k}_4^2{k}_1^8-2k_3^6{k}_1^6-2k_4^6{k}_1^6-k_3^2{k}_4^4{k}_1^6\\ -k_3^4{k}_4^2{k}_1^6+4k_3^8{k}_1^4+4k_4^8{k}_1^4-k_3^2{k}_4^6{k}_1^4-6k_3^4{k}_4^4{k}_1^4-k_3^6{k}_4^2{k}_1^4-2k_3^2{k}_4^8{k}_1^2\\ -k_3^4{k}_4^6{k}_1^2-k_3^6{k}_4^4{k}_1^2-2k_3^8{k}_4^2{k}_1^2+4k_3^4{k}_4^8-2k_3^6{k}_4^6+4k_3^8{k}_4^4]\\ /[4{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_3+k_4\right)}^2\\ \left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{1012}=\frac{\left(2k_1^4-k_3^2{k}_1^2+2k_3^4\right){\left(k_1-k_4\right)}^2{\left(k_3-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)}{32{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}\\ B_{1020}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)}{16{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)}\\ B_{1021}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_3-k_4\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)\left(2k_1^4-k_4^2{k}_1^2+2k_4^4\right)}{32{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}\end{array}$$

$$\begin{array}{l}{B}_{1022}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_1-k_4\right)}^2{\left(k_3-k_4\right)}^4\left(k_1^2-k_4{k}_1+k_4^2\right){\left(k_3^2-k_4{k}_3+k_4^2\right)}^2}{256{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_3+k_4\right)}^4\left(k_1^2+k_4{k}_1+k_4^2\right){\left(k_3^2+k_4{k}_3+k_4^2\right)}^2}\\ B_{1100}=\frac{2k_1^4-k_2^2{k}_1^2+2k_2^4}{2{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)}\end{array}$$

$$\begin{array}{l}{B}_{1101}=[4k_2^4{k}_1^8+4k_4^4{k}_1^8-2k_2^2{k}_4^2{k}_1^8-2k_2^6{k}_1^6-2k_4^6{k}_1^6-k_2^2{k}_4^4{k}_1^6-k_2^4{k}_4^2{k}_1^6\\ +4k_2^8{k}_1^4+4k_4^8{k}_1^4-k_2^2{k}_4^6{k}_1^4-6k_2^4{k}_4^4{k}_1^4-k_2^6{k}_4^2{k}_1^4-2k_2^2{k}_4^8{k}_1^2-k_2^4{k}_4^6{k}_1^2-\\ k_2^6{k}_4^4{k}_1^2-2k_2^8{k}_4^2{k}_1^2+4k_2^4{k}_4^8-2k_2^6{k}_4^6+4k_2^8{k}_4^4]/[4{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)\\ {\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{1102}=\frac{\left(2k_1^4-k_2^2{k}_1^2+2k_2^4\right){\left(k_1-k_4\right)}^2{\left(k_2-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(k_2^2-k_4{k}_2+k_4^2\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)}\\ B_{1120}=\frac{\left(2k_1^4-k_2^2{k}_1^2+2k_2^4\right){\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(k_2^2-k_3{k}_2+k_3^2\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}\end{array}$$

$$\begin{array}{l}{B}_{1121}={\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_3-k_4\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)\\ (4k_2^4{k}_1^8+4k_4^4{k}_1^8-2k_2^2{k}_4^2{k}_1^8-2k_2^6{k}_1^6-2k_4^6{k}_1^6-k_2^2{k}_4^4{k}_1^6-k_2^4{k}_4^2{k}_1^6+4k_2^8{k}_1^4+4k_4^8{k}_1^4+\\ -k_2^2{k}_4^6{k}_1^4-6k_2^4{k}_4^4{k}_1^4-k_2^6{k}_4^2{k}_1^4-2k_2^2{k}_4^8{k}_1^2-k_2^4{k}_4^6{k}_1^2-k_2^6{k}_4^4{k}_1^2-2k_2^8{k}_4^2{k}_1^2+4k_2^4{k}_4^8-2k_2^6{k}_4^6+4k_2^8{k}_4^4)\\ /[64{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_1+k_4\right)}^2\\ {\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{1122}=\left(2k_1^4-k_2^2{k}_1^2+2k_2^4\right){\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_1-k_4\right)}^2\\ {\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^4\left(k_1^2-k_4{k}_1+k_4^2\right)\left(k_2^2-k_4{k}_2+k_4^2\right){\left(k_3^2-k_4{k}_3+k_4^2\right)}^2\\ /[512{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)\\ {\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^4\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right){\left(k_3^2+k_4{k}_3+k_4^2\right)}^2].\end{array}$$

$$\begin{array}{l}{B}_{1200}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right)}{16{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)}\\ B_{1201}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_2-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)\left(2k_1^4-k_4^2{k}_1^2+2k_4^4\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)}\\ B_{1202}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_4\right)}^2{\left(k_2-k_4\right)}^4\left(k_1^2-k_4{k}_1+k_4^2\right){\left(k_2^2-k_4{k}_2+k_4^2\right)}^2}{256{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^4\left(k_1^2+k_4{k}_1+k_4^2\right){\left(k_2^2+k_4{k}_2+k_4^2\right)}^2}\\ B_{1210}=\frac{{\left(k_{1=}-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)\left(2k_1^4-k_3^2{k}_1^2+2k_3^4\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}\end{array}$$

$$\begin{array}{l}{B}_{1211}={\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_2-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)\\ \left(4k_3^4{k}_1^8+4k_4^4{k}_1^8-2k_3^2{k}_4^2{k}_1^8-2k_3^6{k}_1^6-2k_4^6{k}_1^6-k_3^2{k}_4^4{k}_1^6-k_3^4{k}_4^2{k}_1^6+4k_3^8{k}_1^4+4k_4^8{k}_1^4-k_3^2{k}_4^6{k}_1^4\right.\\ \left.-6k_3^4{k}_4^4{k}_1^4-k_3^6{k}_4^2{k}_1^4-2k_3^2{k}_4^8{k}_1^2-k_3^4{k}_4^6{k}_1^2-k_3^6{k}_4^4{k}_1^2-2k_3^8{k}_4^2{k}_1^2+4k_3^4{k}_4^8-2k_3^6{k}_4^6+4k_3^8{k}_4^4\right)\\ /[64{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)\\ {\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{1212}=[{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_2-k_3\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)\left(2k_1^4-k_3^2{k}_1^2+2k_3^4\right)\\ {\left(k_1-k_4\right)}^2{\left(k_2-k_4\right)}^4{\left(k_3-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right){\left(k_2^2-k_4{k}_2+k_4^2\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)]\\ /[512{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)\\ {\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^4{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right){\left(k_2^2+k_4{k}_2+k_4^2\right)}^2\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$B_{1220}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^4\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_2^2-k_3{k}_2+k_3^2\right)}^2}{256{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^4\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_2^2+k_3{k}_2+k_3^2\right)}^2}.$$

$$\begin{array}{l}{B}_{1221}=[{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^4\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_2^2-k_3{k}_2+k_3^2\right)}^2\\ {\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)\left(2k_1^4-k_4^2{k}_1^2+2k_4^4\right)]\\ /[512{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^4\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_2^2+k_3{k}_2+k_3^2\right)}^2\\ {\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{1222}={\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^4\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_2^2-k_3{k}_2+k_3^2\right)}^2\\ {\left(k_1-k_4\right)}^2{\left(k_2-k_4\right)}^4{\left(k_3-k_4\right)}^4\left(k_1^2-k_4{k}_1+k_4^2\right){\left(k_2^2-k_4{k}_2+k_4^2\right)}^2{\left(k_3^2-k_4{k}_3+k_4^2\right)}^2\\ /[4096{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^4\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_2^2+k_3{k}_2+k_3^2\right)}^2\\ {\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^4{\left(k_3+k_4\right)}^4\left(k_1^2+k_4{k}_1+k_4^2\right){\left(k_2^2+k_4{k}_2+k_4^2\right)}^2{\left(k_3^2+k_4{k}_3+k_4^2\right)}^2].\end{array}$$

$$B_{2000}=\frac{1}{16},B_{2001}=\frac{{\left(k_1-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)}{16{\left(k_1+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)},B_{2002}=\frac{{\left(k_1-k_4\right)}^4{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2}{256{\left(k_1+k_4\right)}^4{\left(k_1^2+k_4{k}_1+k_4^2\right)}^2}.$$

$$B_{2010}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)}{16{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)}.$$

$$\begin{array}{l}{B}_{2011}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_1-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(2k_3^4-k_4^2{k}_3^2+2k_4^4\right)}{32{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}.\\ B_{2012}=\frac{{\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_1-k_4\right)}^4{\left(k_3-k_4\right)}^2{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)}{256{\left(k_1+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right){\left(k_1+k_4\right)}^4{\left(k_3+k_4\right)}^2{\left(k_1^2+k_4{k}_1+k_4^2\right)}^2\left(k_3^2+k_4{k}_3+k_4^2\right)}.\\ B_{2020}=\frac{{\left(k_1-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2}{256{\left(k_1+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2}.\\ \begin{array}{l}{B}_{2021}=\frac{{\left(k_1-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2{\left(k_1-k_4\right)}^2{\left(k_3-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)}{256{\left(k_1+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2{\left(k_1+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)}.\\ \begin{array}{l}{B}_{2022}=\frac{{\left(k_1-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2{\left(k_1-k_4\right)}^4{\left(k_3-k_4\right)}^4{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2{\left(k_3^2-k_4{k}_3+k_4^2\right)}^2}{4096{\left(k_1+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2{\left(k_1+k_4\right)}^4{\left(k_3+k_4\right)}^4{\left(k_1^2+k_4{k}_1+k_4^2\right)}^2{\left(k_3^2+k_4{k}_3+k_4^2\right)}^2}.\\ B_{2100}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right)}{16{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right)}.\\ B_{2101}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(2k_2^4-k_4^2{k}_2^2+2k_4^4\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)}.\\ B_{2102}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_4\right)}^4{\left(k_2-k_4\right)}^2{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2\left(k_2^2-k_4{k}_2+k_4^2\right)}{256{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_4\right)}^4{\left(k_2+k_4\right)}^2{\left(k_1^2+k_4{k}_1+k_4^2\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)}.\\ B_{2110}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(2k_2^4-k_3^2{k}_2^2+2k_3^4\right)}{32{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}.\end{array}\end{array}\end{array}$$

$$\begin{array}{l}{B}_{2111}=[{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right){\left(k_1-k_4\right)}^2\\ \left(k_1^2-k_4{k}_1+k_4^2\right)(4k_3^4{k}_2^8+4k_4^4{k}_2^8-2k_3^2{k}_4^2{k}_2^8-2k_3^6{k}_2^6-2k_4^6{k}_2^6-k_3^2{k}_4^4{k}_2^6\\ -k_3^4{k}_4^2{k}_2^6+4k_3^8{k}_2^4+4k_4^8{k}_2^4-k_3^2{k}_4^6{k}_2^4-6k_3^4{k}_4^4{k}_2^4-k_3^6{k}_4^2{k}_2^4-2k_3^2{k}_4^8{k}_2^2\\ -k_3^4{k}_4^6{k}_2^2-k_3^6{k}_4^4{k}_2^2-2k_3^8{k}_4^2{k}_2^2+4k_3^4{k}_4^8-2k_3^6{k}_4^6+4k_3^8{k}_4^4\left)\right]\\ /[64{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\\ \left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\\ \left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{2112}=[{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\\ \left(2k_2^4-k_3^2{k}_2^2+2k_3^4\right){\left(k_1-k_4\right)}^4{\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^2{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2\\ \left(k_2^2-k_4{k}_2+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)]/[512{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^2\\ {\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_1+k_4\right)}^4{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\\ {\left(k_1^2+k_4{k}_1+k_4^2\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$B_{2120}=\frac{{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^2{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2\left(k_2^2-k_3{k}_2+k_3^2\right)}{256{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^4{\left(k_2+k_3\right)}^2{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right)}.$$

$$\begin{array}{l}{B}_{2121}=[{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^2{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2\\ \left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_1-k_4\right)}^2{\left(k_3-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)\\ \left(2k_2^4-k_4^2{k}_2^2+2k_4^4\right)]/[512{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^4{\left(k_2+k_3\right)}^2\\ {\left(k_1^2+k_3{k}_1+k_3^2\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\\ \left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{2122}=[{\left(k_1-k_2\right)}^2\left(k_1^2-k_2{k}_1+k_2^2\right){\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^2{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2\\ \left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_1-k_4\right)}^4{\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^4{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2\\ \left(k_2^2-k_4{k}_2+k_4^2\right){\left(k_3^2-k_4{k}_3+k_4^2\right)}^2]/[4096{\left(k_1+k_2\right)}^2\left(k_1^2+k_2{k}_1+k_2^2\right){\left(k_1+k_3\right)}^4\\ {\left(k_2+k_3\right)}^2{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_1+k_4\right)}^4{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^4\\ {\left(k_1^2+k_4{k}_1+k_4^2\right)}^2\left(k_2^2+k_4{k}_2+k_4^2\right){\left(k_3^2+k_4{k}_3+k_4^2\right)}^2].\end{array}$$

$$B_{2200}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2}{256{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2}.$$

$$\begin{array}{l}{B}_{2201}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_4\right)}^2{\left(k_2-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(k_2^2-k_4{k}_2+k_4^2\right)}{256{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2\left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)}\\ B_{2202}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_4\right)}^4{\left(k_2-k_4\right)}^4{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2{\left(k_2^2-k_4{k}_2+k_4^2\right)}^2}{4096{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_4\right)}^4{\left(k_2+k_4\right)}^4{\left(k_1^2+k_4{k}_1+k_4^2\right)}^2{\left(k_2^2+k_4{k}_2+k_4^2\right)}^2}\\ B_{2210}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\left(k_2^2-k_3{k}_2+k_3^2\right)}{256{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right)}\end{array}$$

$$\begin{array}{l}{B}_{2211}=[{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\\ \left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_1-k_4\right)}^2{\left(k_2-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\left(k_2^2-k_4{k}_2+k_4^2\right)\\ \left(2k_3^4-k_4^2{k}_3^2+2k_4^4\right)]/[512{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^2{\left(k_2+k_3\right)}^2\\ \left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\\ \left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{2212}=[{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^2{\left(k_2-k_3\right)}^2\left(k_1^2-k_3{k}_1+k_3^2\right)\\ \left(k_2^2-k_3{k}_2+k_3^2\right){\left(k_1-k_4\right)}^4{\left(k_2-k_4\right)}^4{\left(k_3-k_4\right)}^2{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2\\ {\left(k_2^2-k_4{k}_2+k_4^2\right)}^2\left(k_3^2-k_4{k}_3+k_4^2\right)]/[4096{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^2\\ {\left(k_2+k_3\right)}^2\left(k_1^2+k_3{k}_1+k_3^2\right)\left(k_2^2+k_3{k}_2+k_3^2\right){\left(k_1+k_4\right)}^4{\left(k_2+k_4\right)}^4{\left(k_3+k_4\right)}^2\\ {\left(k_1^2+k_4{k}_1+k_4^2\right)}^2{\left(k_2^2+k_4{k}_2+k_4^2\right)}^2\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$B_{2220}=\frac{{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2{\left(k_2^2-k_3{k}_2+k_3^2\right)}^2}{4096{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^4{\left(k_2+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2}.$$

$$\begin{array}{l}{B}_{2221}=[{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2\\ {\left(k_2^2-k_3{k}_2+k_3^2\right)}^2{\left(k_1-k_4\right)}^2{\left(k_2-k_4\right)}^2{\left(k_3-k_4\right)}^2\left(k_1^2-k_4{k}_1+k_4^2\right)\\ \left(k_2^2-k_4{k}_2+k_4^2\right)\left(k_3^2-k_4{k}_3+k_4^2\right)]/[4096{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^4\\ {\left(k_2+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2{\left(k_1+k_4\right)}^2{\left(k_2+k_4\right)}^2{\left(k_3+k_4\right)}^2\\ \left(k_1^2+k_4{k}_1+k_4^2\right)\left(k_2^2+k_4{k}_2+k_4^2\right)\left(k_3^2+k_4{k}_3+k_4^2\right)].\end{array}$$

$$\begin{array}{l}{B}_{2222}=[{\left(k_1-k_2\right)}^4{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2{\left(k_1-k_3\right)}^4{\left(k_2-k_3\right)}^4{\left(k_1^2-k_3{k}_1+k_3^2\right)}^2\\ {\left(k_2^2-k_3{k}_2+k_3^2\right)}^2{\left(k_1-k_4\right)}^4{\left(k_2-k_4\right)}^4{\left(k_3-k_4\right)}^4{\left(k_1^2-k_4{k}_1+k_4^2\right)}^2\\ {\left(k_2^2-k_4{k}_2+k_4^2\right)}^2{\left(k_3^2-k_4{k}_3+k_4^2\right)}^2]/[65536{\left(k_1+k_2\right)}^4{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2{\left(k_1+k_3\right)}^4\\ {\left(k_2+k_3\right)}^4{\left(k_1^2+k_3{k}_1+k_3^2\right)}^2{\left(k_2^2+k_3{k}_2+k_3^2\right)}^2{\left(k_1+k_4\right)}^4{\left(k_2+k_4\right)}^4{\left(k_3+k_4\right)}^4\\ {\left(k_1^2+k_4{k}_1+k_4^2\right)}^2{\left(k_2^2+k_4{k}_2+k_4^2\right)}^2{\left(k_3^2+k_4{k}_3+k_4^2\right)}^2].\end{array}$$

For a graphical illustration, see Figure 4.

Figure 4.

Four soliton solution:$\left\{k_1=0.489591,k_2=0.68479,k_3=0.104676,k_4=0.733527\right\}$.

3.6 N-soliton solutions

The N-soliton solutions have the form $u\left(x,t\right)=1/2{\partial }_{\mathit{xx}}logf\left(x,t\right)$, where

$$\left\{\begin{array}{l}f\left(x,t\right)={\sum }_{i_1,i_2,\cdots ,i_N=0}^2{B}_{i_1{i}_2\cdots i_N}\underset{j=1}{\overset{N}{\mathrm{\Pi }}}{z}_j^{i_j},\\ z_j=exp\left(k_{j}x-k_j^{7}x\right)\text{for}\mathrm{any}j\end{array}\right.$$E50

In order to find the unknown coefficients $B_{i_1{i}_2\cdots i_N},$ we define

$$\begin{array}{l}\mathrm{\Psi }\left(z_1,z_2,\cdots ,z_N\right)=u_t+2016u^3{u}_x+630u_x^3+2268u_x{u}_{2x}+504u^2{u}_{3x}\\ +252u_{2x}{u}_{3x}+147u_x{u}_{4x}+42{\mathit{uu}}_{5x}+u_{7x}.\end{array}$$

The number $B_{i_1{i}_2\cdots i_N}$ is obtained from the equation

$$\frac{{\partial }^{i_1+i_2+\cdots +i_N}}{\partial z_1^{i_1}\partial z_2^{i_2}\cdots \partial z_N^{i_n}}\mathrm{\Psi }\left(\mathrm{0,0,0},\dots ,0\right)=0$$E51

4. Bilinearization

Let us consider the case when $d=c/2-b$. The family to be considered is

$$u_t+{\mathit{au}}^3{u}_x+{\mathit{bu}}_x^3+{\mathit{cuu}}_x{u}_{2x}+\left(c/2-b\right)u^2{u}_{3x}+\alpha u_{2x}{u}_{3x}+\beta u_x{u}_{4x}+\gamma {\mathit{uu}}_{5x}+u_{7x}=0.$$E52

Let

$$u\left(x,t\right)=A{\partial }_{\mathit{xx}}f\left(x,t\right).$$E53

Plugging the ansatz in Eq. (53) into Eq. (52) and integrating once with respect to x, taking a zero integration constant gives

$$\begin{array}{l}3f_x\left(8f^4\right(4{\mathit{Af}}_{\mathit{xxx}}^2\left(\mathit{Ab}+5\beta -5\gamma \right){D^2}_x\left(f\cdot f\right)+2f(A\left(\gamma -\beta \right)f_{\mathit{\text{xxxxxx}}}{D^2}_x\left(f\cdot f\right)+2f({D^8}_x\left(f\cdot f\right)+\\ D_{\mathrm{xt}}\left(f\cdot f\right)\left)\right)+\left(A\left(\alpha -\beta +\gamma \right)-140\right){D^4}_x{\left(f\cdot f\right)}^2+\left(A\left(\beta +\gamma \right)-112\right){D^2}_x\left(f\cdot f\right){D^6}_x\left(f\cdot f\right))\\ +{D^2}_x{\left(f\cdot f\right)}^4\left({\mathit{aA}}^3+6A\left(12\alpha +4\mathit{Ab}-2\mathit{Ac}+3\beta +77\gamma \right)-20160\right)\\ +4f^2{D^2}_x{\left(f\cdot f\right)}^2{D^4}_x\left(f\cdot f\right)\left(A^2\left(c-2b\right)-3A\left(4\alpha +\beta +19\gamma \right)+3360\right))\\ -48{\mathit{Af}}^2{f}_x^4{f}_{\mathit{xxx}}\left(4\mathit{Ab}+15\left(\gamma -\beta \right)\right){D^2}_x\left(f\cdot f\right)-24{\mathit{Af}}^2{f}_x^2{f}_{\mathit{xxx}}(3\left(4\mathit{Ab}-15\beta +15\gamma \right){D^2}_x{\left(f\cdot f\right)}^2\\ +10f^2\left(\beta -\gamma \right){D^4}_x\left(f\cdot f\right))+8{\mathit{Af}}_x^3(2f^2\left(\beta -\gamma \right)(f^2\left(20f_{\mathit{xxx}}^2-2{\mathit{ff}}_{\mathit{\text{xxxxxx}}}+{D^6}_x\left(f\cdot f\right)\right)\\ +15{D^2}_x\left(f\cdot f\right){D^4}_x\left(f\cdot f\right))+9\left(3\mathit{Ab}+20\left(\gamma -\beta \right)\right){D^2}_x{\left(f\cdot f\right)}^3)+4{\mathit{Af}}^2\left(\beta -\gamma \right)f_{\mathit{xxx}}(-4f^4\left(20f_{\mathit{xxx}}^2+{D^6}_x\left(f\cdot f\right)\right)\\ +8f^5{f}_{\mathit{\text{xxxxxx}}}-45{D^2}_x{\left(f\cdot f\right)}^3+30f^2{D^2}_x\left(f\cdot f\right){D^4}_x\left(f\cdot f\right))+288{\mathit{Af}}^2\left(\beta -\gamma \right)f_x^6{f}_{\mathit{xxx}}\\ +96{\mathit{Af}}_x^7\left(\mathit{Ab}+12\left(\gamma -\beta \right)\right){D^2}_x\left(f\cdot f\right)+48{\mathit{Af}}_x^5(3\left(2\mathit{Ab}-15\beta +15\gamma \right){D^2}_x{\left(f\cdot f\right)}^2\\ +5f^2\left(\beta -\gamma \right){D^4}_x\left(f\cdot f\right))+288A\left(\gamma -\beta \right)f_x^9=0.\end{array}$$

The choices

$$\left\{\alpha =\frac{5\gamma }{2},\beta =\gamma ,a=\frac{15{\gamma }^3}{784},b=0,c=\frac{15{\gamma }^2}{28}\right\}$$E54

will give the following bilinear form

$$D_{\mathit{xt}}^1\left(f\cdot f\right)+D_x^8\left(f\cdot f\right)=0$$E55

This corresponds to the KdV7 $\left(A=2,\gamma =28\right)$

$$u_t+420u^3{u}_x+420u_x{u}_{2x}+210u^2{u}_{3x}+70u_{2x}{u}_{3x}+28u_x{u}_{4x}+28{\mathit{uu}}_{5x}+u_{7x}=0.$$E56

This KdV7 admits one and two soliton solutions. However, it does not have three solitons solutions despite the fact that it admits bilinear form.

One soliton solution: $u\left(x,t\right)=2{\partial }_{\mathit{xx}}log\left(1+exp\left(k_{1}x-k_1^{7}t\right)\right).$.

Two soliton solution: $u\left(x,t\right)=2{\partial }_{\mathit{xx}}log\left(1+exp\left(k_{1}x-k_1^{7}t\right)+exp\left(k_{2}x-k_2^{7}t\right)+A_{12}exp\left(k_{1}x-k_1^{7}t\right)exp\left(k_{2}x-k_2^{7}t\right)\right),$

where

$$A_{1,2}=\frac{{\left(k_2-k_1\right)}^2{\left(k_1^2-k_2{k}_1+k_2^2\right)}^2}{{\left(k_1+k_2\right)}^2{\left(k_1^2+k_2{k}_1+k_2^2\right)}^2}.$$E57

Breather: $u\left(x,t\right)=2{\partial }_{\mathit{xx}}log\left({\mathit{pe}}^{\mathit{kx}-\mathit{\lambda t}}+{\mathit{qe}}^{\mathit{\lambda t}-\mathit{kx}}+rsin\left(\mathit{\kappa x}-\mathit{\mu t}\right)\right)$, where

$$\begin{array}{c}\lambda =k\left(-7{\kappa }^6+k^6-21{\kappa }^2{k}^4+35{\kappa }^4{k}^2\right).\\ \mu =\kappa \left(-{\kappa }^6+7k^6-35{\kappa }^2{k}^4+21{\kappa }^4{k}^2\right).\\ p=-\frac{{\kappa }^2{r}^2{\left(3{\kappa }^2-k^2\right)}^2}{4k^{2}q{\left({\kappa }^2-3k^2\right)}^2}.\end{array}$$E58

Let us consider a more general than Eq. (56) KdV7

$$\begin{array}{l}{u}_t+420u^3{u}_x+420{\mathit{uu}}_x{u}_{\mathit{xx}}+210u^2{u}_{\mathit{xxx}}+70u_{\mathit{xx}}{u}_{\mathit{xxx}}+28u_x{u}_{\mathit{\text{xxxx}}}\\ +28{\mathit{uu}}_{\mathit{\text{xxxxx}}}+u_{7x}+6{\mathit{puu}}_x+45{\mathit{qu}}^2{u}_x+15{\mathit{qu}}_x{u}_{\mathit{xx}}+{\mathit{pu}}_{\mathit{xxx}}+15{\mathit{quu}}_{\mathit{xxx}}+{\mathit{quu}}_{5x}+u_{7x}=0.\end{array}$$

This KdV7 admits the bilinear form

$$D_{\mathit{xt}}^1\left(f\cdot f\right)+{\mathit{pD}}_x^4\left(f\cdot f\right)+{\mathit{qD}}_x^6\left(f\cdot f\right)+D_x^8\left(f\cdot f\right)=0.$$E59

The one soliton solutions are

$$u\left(x,t\right)=\frac{k^2{e}^{k^{3}t\left(k^4+k^{2}q+p\right)+\mathit{kx}}}{{\left(e^{k^{3}t\left(k^4+k^{2}q+p\right)}+e^{\mathit{kx}}\right)}^2}.$$

The two soliton solutions are

$$u\left(x,t\right)={\partial }_{\mathit{xx}}log\left(1+exp\left(k_{1}x-w_{1}t\right)+exp\left(k_{2}x-w_{2}t\right)+A_{1,2}exp\left(k_{1}x-w_{1}t\right)exp\left(k_{2}x-w_{2}t\right)\right),$$

where $w_1=k_1^{3}p+k_1^{5}q+k_1^7,w_2=k_2^{3}p+k_2^{5}q+k_2^7,$ and

$$A_{1,2}=\frac{{\left(k_1-k_2\right)}^2\left(5k_1^{2}q-5k_2{k}_{1}q+5k_2^{2}q+7k_1^4-14k_2{k}_1^3+21k_2^2{k}_1^2-14k_2^3{k}_1+7k_2^4+3p\right)}{{\left(k_1+k_2\right)}^2\left(5k_1^{2}q+5k_2{k}_{1}q+5k_2^{2}q+7k_1^4+14k_2{k}_1^3+21k_2^2{k}_1^2+14k_2^3{k}_1+7k_2^4+3p\right)}.$$E60

On the other hand, direct calculations show that the KdV7

$$\begin{array}{l}{u}_t+\frac{\left(3\alpha -5\beta \right){\left(2\beta +\gamma \right)}^2}{1176}{u}^3{u}_x+\frac{1}{56}\left(\beta -\gamma \right)\left(2\beta +\gamma \right)u_x^3+\frac{1}{28}\left(2\beta +\gamma \right)\left(2\alpha -3\beta +3\gamma \right){\mathit{uu}}_x{u}_{\mathit{xx}}\\ +\frac{1}{28}\left(2\beta +\gamma \right)\left(\alpha -2\beta +2\gamma \right)u^2{u}_{\mathit{xxx}}+\alpha u_{\mathit{xx}}{u}_{\mathit{xxx}}+\beta u_x{u}_{\mathit{\text{xxxx}}}+\gamma {\mathit{uu}}_{\mathit{\text{xxxxx}}}+u_{\mathit{\text{xxxxxxx}}}=0\end{array}$$

may be written in the following Hirota’s bilinear form [4]:

$$\left\{\begin{array}{l}2D_{\mathit{xt}}\left(f\cdot f\right)+\frac{7\left(\beta -\gamma \right)}{2\beta +\gamma }{D}_x^4\left(f\cdot g\right)-\frac{3\left(\beta -3\gamma \right)}{2\beta +\gamma }{D}_x^4\left(f\cdot f\right)+\frac{14\left(6\alpha -8\beta -7\gamma \right)}{2\beta +\gamma }{D}_x^8\left(f\cdot f\right)\\ +\frac{14\left(6\alpha -8\beta -7\gamma \right)}{2\beta +\gamma }\left(g\cdot g\right)=0.\\ D_x^4\left(f\cdot f\right)-\left(f\cdot g\right)=0.\\ u\left(x,t\right)=A{\partial }_{\mathit{xx}}logf\left(x,t\right),A=\frac{168}{\gamma +2\beta }.\end{array}\right.$$E61

The seventh-order Kaup-Kuperschmidt Eq. (4) belongs to this class $\left(A=1/2\right)$. Using the obtained bilinear form, we may obtain all the results we presented in previous sections (for the special case $d=c/2-b$).

5. Forced KdV7

The forced KdV7 is written as

$$u_t+{\mathit{au}}^3{u}_x+{\mathit{bu}}_x^3+{\mathit{cuu}}_x{u}_{2x}+{\mathit{du}}^2{u}_{3x}+\alpha u_{2x}{u}_{3x}+\beta u_x{u}_{4x}+\gamma u_{5x}+u_{7x}=f\left(t\right).$$E62

The forced Sawada-Kotera-Ito Eq. (2) and the forced Lax Eq. (3) admit the exact solution.

$$u_t+252u^3{u}_x+63u_x^3+378u_x{u}_{2x}+126u^2{u}_{3x}+63u_{2x}{u}_{3x}+42u_x{u}_{4x}+21{\mathit{uu}}_{5x}+u_{7x}=f\left(t\right).$$

Exact solution:

$$u\left(x,t\right)=B+2{sech}^2\left(x-\lambda \left(t\right)\right)+F\left(t\right),$$E63

where

$$\begin{array}{l}\lambda \left(t\right)=4\int \left(63B^3+189B^{2}F\left(t\right)+126B^2+189\mathit{BF}{\left(t\right)}^2+252\mathit{BF}\left(t\right)+84B+63F{\left(t\right)}^3+126F{\left(t\right)}^2+84F\left(t\right)+16\right)\mathit{dt}.\end{array}$$E64

and

$$F\left(t\right)=\int f\left(t\right).$$E65

$$u_t+140u^3{u}_x+70u_x^3+280u_x{u}_{2x}+70u^2{u}_{3x}+70u_{2x}{u}_{3x}+42u_x{u}_{4x}+14{\mathit{uu}}_{5x}+u_{7x}=0.$$

• Exact solution:

$$u\left(x,t\right)=B+2{sech}^2\left(x-\lambda \left(t\right)\right)+F\left(t\right),$$E66

where

$$\lambda \left(t\right)=4\int \left(\begin{array}{c}35B^3+105B^{2}F\left(t\right)+70B^2+105\mathit{BF}{\left(t\right)}^2+140\mathit{BF}\left(t\right)\\ +56B+35F{\left(t\right)}^3+70F{\left(t\right)}^2+56F\left(t\right)+16\end{array}\right)\mathit{dt}.$$E67

and

$$F\left(t\right)=\int f\left(t\right).$$E68

For other parameter values, we obtained the following result:

If $\alpha +\beta +\gamma \ne 0$ and

$$\begin{array}{c}a=\frac{1}{63}d\left(\alpha +\beta +\gamma \right)\\ b=\frac{1}{126}\left(-{\alpha }^2+4\mathit{\alpha \beta }-11\mathit{\alpha \gamma }+5{\beta }^2-5\mathit{\beta \gamma }-10{\gamma }^2+126d\right)\\ c=\frac{1}{21}\left(5\mathit{\alpha \gamma }+5\mathit{\beta \gamma }+5{\gamma }^2-42d\right)\end{array}$$E69

then the forced KdV7 in Eq. (2) admits the exact solution

$$u\left(x,t\right)=B+\frac{252}{\alpha +\beta +\gamma }{sech}^2\left(x-\lambda \left(t\right)\right)+F\left(t\right),$$E70

where

$$\begin{array}{l}\lambda \left(t\right)=\frac{1}{63}\int \left(\begin{array}{c}F\left(t\right)\left(3\alpha B^{2}d+3\beta B^{2}d+3B^2\mathit{\gamma d}+504\mathit{Bd}+1008\gamma \right)\\ +F{\left(t\right)}^2\left(3\mathit{\alpha Bd}+3\mathit{\beta Bd}+3\mathit{B\gamma d}+252d\right)+F{\left(t\right)}^3\left(\mathit{\alpha d}+\mathit{\beta d}+\mathit{\gamma d}\right)\end{array}\right)\mathit{dt}\\ +\frac{1}{63}t\left(\alpha B^{3}d+\beta B^{3}d+B^3\mathit{\gamma d}+252B^{2}d+1008\mathit{B\gamma }+4032\right),\\ F\left(t\right)=\int f\left(t\right).\end{array}$$

See also [5].