Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations

It is well known that the wave propagation depends mainly on its velocity and frequency in one direction for a single wave[1-3]. There are many literatures devoted researches of single wave propagation such as solitary wave, periodic wave, chirped wave, rational wave etc [4-6]. However, what can be happen when two and even more waves with different features propagate together along different directions? In the past decades, many methods have been proposed for seeking two waves and multi-wave solutions to nonlinear models in modern physics. Recently, some effective and straight methods have been proposed such as homoclinic test approach(HTA)[7-8], extended homoclinic test approach(EHTA)[9-10] and three wave method [11-12]. These methods were applied to many nonlinear models. Several exact waves with different properties have been found out, such as periodic solitary wave, breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave, kinky kinkwave, periodic kinkwave, two-solitarywave, doubly periodicwave, doubly breather solitary wave, and so on. Because of interaction between waves with different features in propagation process of two-wave or multi-wave, some new phenomena have been discovered and numerically simulated, for example, resonance and non resonance, fission and fusion, bifurcation and deflexion etc. Furthermore, similar to the bifurcation theory of differential dynamical system, constant equilibrium solution of nonlinear evolution equation and propagation velocity of a wave as parameters are introduced to original equation, and then by using the small perturbation of parameter at a special value, two-wave or multi-wave propagation occurs new spatiotemporal change such as bifurcation of breather multi-soliton, periodic bifurcation and soliton degeneracy and so on.


Introduction
It is well known that the wave propagation depends mainly on its velocity and frequency in one direction for a single wave [1][2][3]. There are many literatures devoted researches of single wave propagation such as solitary wave, periodic wave, chirped wave, rational wave etc [4][5][6]. However, what can be happen when two and even more waves with different features propagate together along different directions? In the past decades, many methods have been proposed for seeking two waves and multi-wave solutions to nonlinear models in modern physics. Recently, some effective and straight methods have been proposed such as homoclinic test approach(HTA) [7][8], extended homoclinic test approach(EHTA) [9][10] and three wave method [11][12]. These methods were applied to many nonlinear models. Several exact waves with different properties have been found out, such as periodic solitary wave, breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave, kinky kink wave, periodic kink wave, two-solitary wave, doubly periodic wave, doubly breather solitary wave, and so on. Because of interaction between waves with different features in propagation process of two-wave or multi-wave, some new phenomena have been discovered and numerically simulated, for example, resonance and non resonance, fission and fusion, bifurcation and deflexion etc. Furthermore, similar to the bifurcation theory of differential dynamical system, constant equilibrium solution of nonlinear evolution equation and propagation velocity of a wave as parameters are introduced to original equation, and then by using the small perturbation of parameter at a special value, two-wave or multi-wave propagation occurs new spatiotemporal change such as bifurcation of breather multi-soliton, periodic bifurcation and soliton degeneracy and so on.
This chapter mainly focus on explanation of different test methods and comprehensive applications to two-wave or multi-wave propagation. New methods will be described such as HTA, EHTA, Three-wave method and parameter small perturbation method. The spatiotemporal variety in exact two-wave and multi-wave propagation will be investigated and numerically simulated. In this chapter, some important models such as shallow water wave propagation models under the transverse long-wave disturbance Potential Kadomtsev-Petviashvili equation [13][14][15][16][17], Kadomtsev-Petviashvili [18][19][20][21][22][23][24][25][26][27] equation, and specially, Kadomtsev-Petviashvili with positive dispersion equation are considered. Applied new methods to these equations, some new two-wave and multi-wave are obtained and spatiotemporal variety in multi-wave propagation is investigated and numerically simulated.
This chapter consists of four sections. In section 1, we introduce some methods including homoclinic test approach (HTA), extended homoclinic test approach(EHTA), three-wave method and parameter small perturbation method.
In section 2, we consider the potential Kadomtsev-Petviashvili equation and investigate the exact periodic kink-wave and degeneracy of soliton. In section 3, we consider the Kadomtsev-Petviashvili equation and investigate periodic bifurcation, deflexion of two solitary waves. In section 4, we consider the Kadomtsev-Petviashvili equation with positive dispersion. By using three-wave method, we obtain some breather kinds of multi-solitary wave solutions, and investigate the fission and fusion of multi-wave.

Homoclinic test approach
Consider a (2+1) dimensional nonlinear evolution equation of the general form where F is a polynomial of u(x, y, t) and its derivatives, t represents time variable and x, y represent spatial variables. Assume that there exists a transformation of unknown function such that Eq.(1) becomes a bilinear equation in the following form where G is a general polynomial in D t , D x , D y , · · · , where the Hirota's bilinear operator D-operator is defined by [2] D m Traditionally, one obtains two solition solutions with the assumption where d is a constant, η 1 = k 1 x + l 1 y + c 1 t, η 2 = k 2 x + l 2 y + c 2 t, and k j , l j , c j , j = 1, 2 are real numbers. If we treat k 1 and k 2 as complex numbers by taking k 1 = −k 2 = ik, i 2 = −1 and assuming l 1 = l 2 = l and c 1 = c 2 = c, Eq.(4) can be convert into the following form In order to get more forms solution of Eq.(1), we use a more general Ansätz which contains complete variables of Eq.(2) replacing Eq.(5): where where all of d, k j , l j , c j , j = 1, 2 may be real numbers or complex numbers. This process is similar to the procedure that one can obtain the homoclinic orbit of the defocusing nonlinear Schrödinger equation from its dark-hole soliton solutions [1]. As a result, we call this skill as "Homoclinic test approach".
To derive analytic expression, we can take the following procedure in detail: inserting Eq.(6) into Eq.(2), then equating the coefficients of the same kind terms to zero, subsequently, solving the resulting algebraic equations to determine the relationship between variables k j , l j , j = 1, 2, · · · with the help of symbolic computation software such as Maple. In Eq. (7), noting the cos functions is meaningful because we often take into account periodic effect in real physical background. Indeed, we can observe this solution is periodic breathing from their profile.

Extended homoclinic test approach
After substituting Eq.(6) into Eq. (2), we can get that whether the Eq.(2) has the nonzero solution. Furthermore, we do some mathematical simplicity. Rewrite Eq.(6) as follows: where b 1 , d are constants. Now, factoring out the exponentials produces: Exploiting this Ansätz to obtain the new solutions of nonlinear evolution equation is called "Extended Homoclinic test approach". To derive analytic expression, we can take the following procedure in detail: inserting Eq.(10) into Eq.(2), then equating the coefficients of the same kind terms to zero, subsequently, solving the resulting algebraic equations to determine the relationship between variables k j , l j , j = 1, 2, · · · with the help of symbolic computation software such as Maple.

Three-wave method
Multi-wave solutions are important because they reveal the interactions between the inner-waves and the various frequency and velocity components. The whole multi-wave solution, for instance, may sometimes be converted into a single soliton of very high energy that propagates over large regions of space without dispersing and an extremely destructive wave is therefore produced of which the tsunami is a good example. Since all double-wave solutions can be found by using the exp-function method proposed by Fu and Dai [18], we propose an extension of the three-soliton method [6] in this section (called the three-wave method) for finding coupled wave solutions. Consider a high dimensional nonlinear evolution equation of the general form where u = u(x, y, z, t) and F is a polynomial u of and its derivatives, t represents time variable and x, y, z represent spatial variables. The three-wave method operates as follows.
Step 1: By Painleve analysis, a transformation is made for some new and unknown function f .
Step 3: Traditionally, we taking the following Ansätz to obtain the three soliton solution f = 1 + e ξ 1 + e ξ 2 + e ξ 3 + a 12 e ξ 1 +ξ 2 + a 13 e ξ 1 +ξ 3 + a 23 e ξ 2 +ξ 3 + a 123 e ξ 1 +ξ 2 +ξ 3 (14) where a 13 , a 23 , a 123 are the constants. Eq. (14) can be rewritten as where Thus, this three soliton Ansätz contains four variables η 1 , η 2 , η 3 and η 4 . Here, we treat it in a different way. We factor out the e η 1 2 in Eq. (16) and decrease the numbers of variables to three terms. On the other hand, we set some parameters in a complex way. At last, the above analysis allows us to construct the following assumptions: or f = e −ξ 1 + δ 1 cos(ξ 2 ) + δ 2 sinh(ξ 3 ) + δ 3 e ξ 1 (19) where δ j , j = 1, 2, 3 are constants. In fact, from Eq. (18) or Eq. (19), it is easily seen that it only contains three wave variables. As a result, we call this method "three wave method". It is obvious that three-wave method is the extension and improvement of the traditional three-soliton method.
Step 5: Solve the set of algebraic equations in Step 4 using Maple and solve for a j , b j , c j , d j and δ j , j = 1, 2, 3.

Introducing parameters and small perturbation method
In this section, we still consider a high dimensional nonlinear evolution equation of the general form where u = u(x, y, z, t) and F is a polynomial u of and its derivatives, t represents time variable and x, y, z represent spatial variables. Here, we introduce a parameter to Eq. (20) in two ways.
(i). Initial solution as a parameter is introduced to Eq. (20). We assume that the u 0 is an initial solution of Eq. (20), then we use the transformation to convert nonlinear evolution equation (20) into Hirota bilinear form which contains the small perturbation parameters u 0 : where D is also the Hirota D-operator. The next step is to exploit the existent method to solving the Eq. (22). The perturbation parameter u 0 plays an important role to the resulting solution, where the spatiotemporal feature in multi-wave propagation including velocity and direction even the shape will change as u 0 makes small perturbation.
(ii). The velocity of a wave variable as a parameter is introduced to Eq. (20). We can assume that ξ = z − αt in Eq. (20), where ξ is a new wave variable and α is its velocity. Then Eq. (20) can be became to Then solving Eq. (23). In this case, the spatiotemporal feature in multi-wave propagation for Eq.(20) may happen outstanding change as α makes a small perturbation.

Potential Kadomtsev-Petviashvili equation
Potential Kadomtsev-Petviashvili (PKP) equation studied in this section is described as It is well known that the PKP equation arose in number of remarkable non-linear problems both in physics and mathematics, the solutions of PKP equation have been studied extensively in various aspects. By applying various methods and techniques to PKP equation, exact traveling wave solutions, linearly solitary wave solutions, soliton-like solutions and some numerical solutions have been obtained [1][2][3][5][6][7]. Recently, two soliton and periodic soliton solutions were presented, resonance and non-resonance interactions between periodic soliton and different line solitons were investigated [6].
In this section, we use homoclinic test method and extended homoclinic test technique to seek periodic soliton solution, exact periodic kink-wave solution, periodic soliton solution and doubly periodic solution of PKP equation. Furthermore, it is explicitly exhibited that the feature of solution is different varying with direction of wave propagation on the x -axis, periodic soliton is degenerated into doubly periodic wave when the direction of a wave propagation changes from progressing to the left into right.

Degenerative periodic solitary wave
In this section, the periodic solition solution is constructed by homoclinic test technique and bilinear form method and the degeneracy of solitary wave is investigated.
where α is a wave velocity. Let u = (ln F) ξ , then Eq.(25) can be reduced into the following bilinear form . With regard to Eq.(26), using the homoclinic test technique, we can seek the solution in the form where A, p, Ω, γ, b 1 and b 2 are all real to be determined below.
Substituting Eq. (27) into Eq. (26) yields the exact solution of Eq.(24) in the form where parameters A, b 1 , b 2 , Ω, p and γ satisfy dispersive relations the exact solution to PKP equation can be expressed as Here, we choice 0 < p 2 < − α 4 such that b 2 1 > 0, then Eq.(30) is the periodic soliton solution of PKP equation which is a periodic traveling wave progressing to the left with velocity |α| on the x-axis, and meanwhile is a soliton on the y-axis (see Fig.1).
Making a variable transformation ξ = x − αt, η = iy in Eq. (24), then it can be transformed into the following form Letting u = (ln F) ξ , being similar to the way of dealing with PKP equation in above, we take We can see that Ω 2 1 ≥ 0 always holds for every real p 1 as long as α ≥ 0. Taking ξ = x − αt, η = iy in Eq.(33), the exact solution to PKP equation (α ≥ 0) is expressed as In particularly, taking Obviously, both cos p 1 (x − αt) and cos(Ω 1 y) are periodic, so the solution given by Eq.(37) is an doubly periodic solution which is a periodic traveling wave progressing to the right with velocity α on the x-axis, and meanwhile is a periodic standing wave on the y-axis (see Fig.2).
It is important that we can take the same p and p 1 such that p 2 = p 2 1 = | α k | = p 2 0 in Eq.(30) and Eq.(37) from the constraint condition Eq.(31) and Eq.(36) respectively, where k > 4 is an arbitrary real number. Therefore, we obtain a periodic soliton solution and an doubly periodic solution which have the same period with x-direction as follows It is easy to find that the feature of solution of Eq. (24) is different from Eq.(38) on the arbitrary small neighborhood of velocity α = 0. There exists a periodic soliton solution which is a periodic with x-direction, and a soliton with y-direction as well as α < 0. When α ≥ 0, this periodic soliton degenerates into a doubly periodic wave which are periodic with both x-direction and y-direction. However, period with x-direction is preserved identically.

Exact periodic kink-wave solution
In this section, a new type of periodic kink-wave solution for PKP equation is obtained using extended homoclinic test technique.
Taking a transformation u = (ln F) x Then Eq. (24) can be transformed as In this case, we let Following the procedure of extended homoclinic test technique, we derive the periodic kink-wave solution of PKP equation(see Fig.3) It can be rewritten as Specially, taking b 1 = 2 √ 5k and γ = ln k, k > 1, yields a periodic kink solution as follows It shows a periodic kink-wave whose speed is 11p 2 and period is 2π/3p of space variable y. It exhibits elastic interaction between a solitary wave and periodic wave with the same speed in opposed direction each other. It is an interesting phenomenon in fluid mechanics (see Fig.3).

Kadomtsev-Petviashvili equation
The (2+1)-dimensional (two spatial and one temporal) generalization of Korteweg-de Vries equation (KdV) was given by Kadomtsev and Petviashvili to discuss the stability of (1+1)-dimensional soliton to the transverse long-wave disturbances, which is known as KP equation or the (2+1)-dimensional KdV equation. There are two distinct versions of KP equation, which can be written in normalized form as follows [4]: The propagation property of solitons depends essentially on the sign of s 2 in equation. The coefficient is defined as follows: s = ±i, i 2 = −1 for negative dispersion and s = ±1 for positive dispersion. Here u = u(x, y, t) is a scalar function, x and y are respectively the longitudinal and transverse spatial coordinates, the subscripts x, y, t denote partial derivatives. When s = ±i, it is usually called KPI, while for s = ±1, it is called KPII. KP equation is the natural generalization of the well known KdV equation (u t − 6uu x − u xxx = 0) from one to two spatial dimensions. It arises naturally in many other applications, particularly in plasma physics, gas dynamics, and elsewhere. Both KPI and KPII are exactly integrable via the Inverse Scattering Transformation. Kadomtsev and Petviashvili have shown that the line soliton of the KP equation is unstable in the case of positive dispersion and is stable for the negative dispersion [3,22]. The solutions of KP equation have been studied extensively in various aspects. The inclined periodic soliton solution and the lattice soliton solution were expressed as exact imbricate series of rational soliton solutions to KP equation with positive dispersion [22]. Resonant interaction of two-soliton among three obliquely oriented solitons in higher dimension was first studied by Miles [26]. Y.Kodama and Dai have proved that the KP equation provides line solitons in shallow water and these solitons can be of resonance [7,27].
In this section, spatial-temporal bifurcation for KP equation is considered, several types of exact solutions to KP equation were constructed by bilinear form and homoclinic test approach. It is explicitly analyzed that the feature of the solitary wave is different on the both sides of equilibrium solution u 0 = − 1 6 , which is a unique periodic bifurcation point for KPI and deflexion point of soliton for KPII. As for KPI, when the equilibrium u 0 varies from one side of − 1 6 to another side, two-solitary wave changes into doubly periodic wave. Whereas, the y-periodic solitary wave changes into x − t-periodic solitary wave for KPII, the propagation direction of periodic solitary wave occurs outstanding deflexion. In addition, some new type of multi-wave solutions are obtained using three-wave method.

Kadomtsev-Petviashvili (KP) equation in normalized variable u(x, y, t) reads
It is easily to note that u = u 0 is an equilibrium solution of KP equation, where u 0 is an arbitrary constant. Now, we consider KPI equation Using "homoclinic test technique", we are going to seek the solution of the form where parameters satisfy It is obviously that u 0 < − 1 6 is required so that the conditions Ω 2 > 0, b 2 1 > 0 and p 2 < − 1+6u 0 4 can be satisfied and u 0 is a free parameter.

KPII equation is given by
Making a variable transformation ξ = x − t in Eq.(54), it can be transformed into the following form Letting u = u 0 + 2(ln F) ξξ and using a similar way dealing with KPI, we take By computing, the exact solution of Eq.(55) is given by where parameters satisfy It is easily to see that u 0 ≥ − 1 6 is available as long as p 2 ≥ 1 + 6u 0 . Taking ξ = x − t into Eq.(56), the exact solution to KPII equation is expressed as Obviously, cos p(x − t) is periodic, so the solution given by Eq.(58) is a periodic soliton solution with x − t-direction (see Fig.5). Comparing Eq.(47) with Eq.(55), it is easily to find that the Eq.(47) may be changed into Eq.(55) and vice versa by using the temporal and spatial transformation (ξ, y) → (iξ, iy). Because the solutions of KP equation are real functions, it is naturally to take specially b 2 = 1, γ = 0. Making variable transformation ξ → iξ, y → iy, i 2 = −1 in Eq.(50) and Eq.(56) yields It is noted that the solution given by Eq.(59) is a singular periodic solution to KPI equation. In order to avoid the singularity, we set cos(p(x − t)) > 0 and cos(Ωy) > 0 (see Fig.6).
Besides, the y-periodic soliton solution to KPII is also given by where The solution given by Eq.(60) represents periodic soliton with y-direction (see Fig.7). According to above discussion, we get that u 0 = − 1 6 is a unique periodic bifurcation point for KPI and deflexion of soliton for KPII. Around the both sides at u 0 , the property of solutions to KPI and KPII is all changed. As for KPI, when the equilibrium u 0 varies from one side of − 1 6 to another side, two-soliton solution changes into doubly periodic solution. Whereas, the y-periodic soliton changes into x-periodic soliton for KPII. The double-soliton waves and doubly periodic soliton waves of KPI, periodic soliton waves on different spatial variable of KPII are interchanged around u 0 .
Similarity, the breather two-solitary wave solutions of KPII equation also obtained as follows Here a −1 , a 2 , p 1 and β 2 are free parameters. And a −1 , a 2 , p 1 and p 3 are free parameters.

Kadomtsev-Petviashvili equation with positive dispersion
The purpose of this section is to investigate the fission and fusion interactions of the breather-type multi-solitary waves solutions to the KP equation with positive dispersion.
By transformation of independent variable t → −t, y → √ 3y in (*), KP equation with positive dispersion can be written as By the transformation of a dependent variable u Eq.(66) can be transformed into the bilinear form where f (x, y, t) is a real function.

Fission and fusion of multi-wave
In the following, by using generalized three-wave type of Ansätz approach, we study the interaction and spatiotemporal feature of three-wave of KP equation with positive dispersion. Now we suppose the solution of Eq.(68) as where 3 and a j , b j , c j , δ j , j = 1, 2, 3 are some constants to be determined. Substituting the Ansätz Eq.(69) into Eq. (68) will produce the following relations: Fig.8 is the process of interaction for two solitons solutions with the evolution of time, where s 1 and s 2 are the different two solitons, and s 12 represents the interaction between s 1 , s 2 and the periodic wave cos(ξ 2 ). The phenomena of soliton interaction are clearly presented. It shows that the two soliton experience interaction, they will fusion with few oscillations and later travel ahead continuously. The value of δ 2 will determine the length of resonant soliton. Obviously, the solitons s 1 and s 2 can not approach each together closely from picture. They interact produce the breather wave.

Conclusion
Using Homoclinic test approach, Extend Homoclinic test approach, Three-wave method and Introducing parameters and small perturbation method, we obtain novel solutions of Potential Kadomtsev-Petviashvili equation and Kadomtsev-Petviashvili equation such as periodic solitary wave,breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave,kinky kink wave, periodic kink wave, two-solitary wave, doubly periodic wave, doubly breather solitary wave. Moreover, we observed that there were differently spatiotemporal features in two-wave and multi-wave propagations including the degeneracy of soliton, periodic bifurcation and soliton deflexion of two-wave, fission and fusion of breather two-wave and so on. In future, we intend to study the stability and the interactions patterns of N-wave solutions in KP equation. What's more, can we obtain similar results to another integrable or non-integrable system? How can one use the soliton fission and fusion of models to study the practically observed soliton fission and fusion in the experiments?