Open access peer-reviewed chapter

Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 2

Written By

Zhou Guo-Quan

Submitted: July 2nd, 2020 Reviewed: July 22nd, 2020 Published: September 22nd, 2020

DOI: 10.5772/intechopen.93450

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A revised and rigorously proved inverse scattering transform (IST for brevity) for DNLS+ equation, with a constant nonvanishing boundary condition (NVBC) and normal group velocity dispersion, is proposed by introducing a suitable affine parameter in the Zakharov-Shabat IST integral; the explicit breather-type and pure N-soliton solutions had been derived by some algebra techniques. On the other hand, DNLS equation with a non-vanishing background of harmonic plane wave is also solved by means of Hirota’s bilinear formalism. Its space periodic solutions are determined, and its rogue wave solution is derived as a long-wave limit of this space periodic solution.


  • soliton
  • nonlinear equation
  • derivative nonlinear Schrödinger equation
  • inverse scattering transform
  • Zakharov-Shabat equation
  • Hirota’s bilinear derivative method
  • DNLS equation
  • space periodic solution
  • rogue wave

1. Breather-type and pure N-soliton solution to DNLS+ equation with NVBC based on revised IST

DNLS+ equation with NVBC, the concerned theme of this section, is only a transformed version of modified nonlinear Schrödinger equation with normal group velocity dispersion and a nonlinear self-steepen term and can be expressed as


here the subscripts represent partial derivatives.

Some progress have been made by several researchers to solve the DNLS equation for DNLS equation with NVBC, many heuristic and interesting results have been attained [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. An early proposed IST worked on the Riemann sheets can only determine the modulus of the one-soliton solution [3, 15]. References [4, 5, 16] had attained a pure single dark/bright soliton solution. Reference [6] had derived a formula for N soliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation; but just as reference [7, 9] pointed out, this multi-soliton solution is still difficult to exhibit the internal elastic collisions among solitons and the typical asymptotic behaviors of multi-soliton of DNLS equation. By introducing an affine parameter in the integral of Zakharov-Shabat IST, reference [7] had found their pure N soliton solution for a special case that all the simple poles (zeros of aλ) were located on a circle of radius ρ centered at the origin, while reference [8] also found its multi-soliton solution for some extended case with N poles on a circle and M poles out of the circle, and further developed its perturbation theory based on IST. Reference [7] constructed their theory by introducing an damping factor in the integral of Zakharov-Shabat IST, to make it convergent, and further adopted a good idea of introducing an affine parameter to avoid the trouble of multi-value problem in Riemann sheets, but both of their results are assumed N soliton solutions and the soliton solution gotten from their IST had a self-dependent and complicated phase factor [7, 8, 9], hence reference [8] had to verify an identity demanded by the standard form of a soliton solution (see expression (52) in Ref. [8]). Such kind of an identity is rather difficult to prove for N2 case even by the use of computer techniques and Mathematica. On the other hand, author of reference [8] also admits his soliton solution is short of a rigorous verification of standard form. Then questions naturally generates – whether the traditional IST for DNLS equation with NVBC can be avoided and further improved? And whether a rigorous manifestation of soliton standard form can be given and a more reasonable and natural IST can be constructed?

A newly revised IST is thus proposed in this section to avoid the dual difficulty and the excessive complexity. An additional affine factor 1/λ, λ=z+ρ2z1/2, is introduced in the Z-S IST integral to make the contour integral convergent in the big circle [7, 10, 11, 12, 13]. Meanwhile, the additional two poles on the imaginary axis caused by λ=0 are removable poles due to the fact that the first Lax operator Lλ0, as λ0. What is more different from reference [7] is that we locate the N simple poles off the circle of radius ρ centered at origin O, which corresponds to the general case of N breather-type solitons. When part of the poles approach the circle, the corresponding part of the breathers must tend to the pure solitons, which is the case described in Ref. [8]. The resulted one soliton solution can naturally tend to the well-established conclusion of VBC case as ρ0 [17, 18, 19, 20] and the pure one soliton solution in the degenerate case. The result of this section appears to be strict and reliable.

1.1 The fundamental concepts for the IST theory of DNLS equation

Under a Galileo transformation xtx+ρ2tt, DNLS +Eq. (1) can be changed into


with nonvanishing boundary condition:


According to references [7, 8, 9], the phase difference between the two infinite ends should be zero. The Lax pair of DNLS+Eq. (3) is


where σ3, σ+, and σ involve in standard Pauli’s matrices and their linear combination. Here and hereafter a bar over a variable represents complex conjugate. An affine parameter z and two aided parameters η, λ are introduced to avoid the trouble of dealing with double-valued functions on Riemann sheets


The Jost functions satisfy first Lax equation


here Jost functions FxzΨxzΦ(xz).


The free Jost function Exz can be easily attained as follows:


which can be verified satisfying Eq. (7). The monodramy matrix is


which is defined by


Some useful and important symmetry properties can be found


Symmetry relations in (13) lead to


The above symmetry relations further result in


Other important symmetry properties called reduction relations can also be easily found


The above symmetry properties lead to following reduction relations among Jost functions


The important symmetries among the transition coefficients further resulted from (12), (21), and (22):


On the other hand, the simple poles, or zeros of az, appear in quadruplet, and can be designated by zn, (n=1,2,,2N), in the I quadrant, and zn+2N=zn in the III quadrant. Due to symmetry (17), (18) and (23), the n’ th subset of zero points is


And we arrange the 2N zeros in the first quadrant in following sequence


According to the standard procedure [21, 22], the discrete part of az can be deduced


At the zeros of az, or poles zn, n=122N12N, we have


Using symmetry relation in (14), (17), (21)(24), (27), we can prove that


2. Relation between the solution and Jost functions of DNLS+ equation

The asymptotic behaviors of the Jost solutions in the limit of λ can be obtained by simple derivation. Let F·1=ψ˜xλ, then Eq. (7) can be rewritten as


then we have


We assume a function g to satisfy the following equation


Substituting (31)(32) into Eq. (30), we have


In the limit of z, gx can be expanded as sum of series of z2j, j=0,1,2,




Inserting formula (34) and (35) in Eq. (33) at z leads to


Then we find a useful formula


which expresses the conjugate of the solution u in terms of Jost functions as z.

2.1 Introduction of time evolution factor

In order to make the Jost functions satisfy the second Lax equation, a time evolution factor htz should be introduced by a standard procedure [21, 22] in the Jost functions and the scattering data. Considering the asymptotic behavior of the second Lax operator


we let htk to satisfy




Due to


from (39) and (40), we have


Therefore, the complete Jost functions should depend on time as follows


Nevertheless, hereafter the time variable in Jost functions will be suppressed because it has no influence on the treatment of Z-S equation. By a similar procedure [9, 15], the scattering data has following time dependences


2.2 Zakharov-Shabat equations and breather-type N-soliton solution

A 2 × 1 column function Πxz is introduced as usual


here and hereafter note “” represents definition. There is an abrupt jump for Πxz across both real and imaginary axes




is called the reflection coefficient. Due to μ00 in (34), Jost solutions do not tend to the free Jost solutions Exz in the limit of z. This is their most typical property which means that the usual procedure of constructing the equation of IST by a Cauchy contour integral must be invalid. In view of these abortive experiences, we proposed a revised method to derive a suitable IST and the corresponding Z-S equation by multiplying an inverse spectral parameter 1/λ,λ=z+ρ2z1/2, before the Z-S integrand. Meanwhile, our modification produces no new poles since the Lax operator Lλ0, as λ0. In another word, the both additional poles z0=± generated by λ=0 are removable. Under reflectionless case, that is, rz=0, the Cauchy integral along with contour Γ shown in Figure 1 gives

Figure 1.

The integral path for IST of the DNLS+.






Note that (27), (45), and (46) have been used in (48). The minus sign before the sum of residue number in (48) comes from the clock-wise contour integrals around the 4 N simple poles when the residue theorem is used, shown in Figure 1. By a standard procedure, the time dependences of bn and cn similar to (43) can be derived


In the reflectionless case, the Zakharov-Shabat equations for DNLS+ equation can be derived immediately from (48) as follows


here Λλη,Λnλznηzn=λnηn, and in Eqs. (52) and (53), the terms corresponding to poles zn, (n=1,2,,2N), have been combined with the terms corresponding to poles zn+2N=zn. Substituting Eqs. (52) and (53) into formula (36) and letting z, we attain the conjugate of the raw N-soliton solution (the time dependence is suppressed).


Letting z=ρ2zm1, (m=1,2,,2N), respectively, in Eqs. (52) and (53), by use of reduction relations (19), (21), and (22), we can further change Eqs. (52) and (53) into the following form


m=1,2,,2N. An implicit time dependence of the complete Jost functions ψ1 and ψ2 besides cn should be understood. To solve Eq. (58), we define that


BMatrixBnm2N×2N, with


Then Eq. (58) can be rewritten as


or in a more compact form


The above equation gives


Note that the choice of poles, zn,n=12N, should make detI+B nonzero and I+B an invertible matrix. On the other hand, Eq. (55) can be rewritten as


hereafter a superscript “T” represents transposing of a matrix. Substituting Eq. (65) into (66) leads to






To solve Eq. (57), we define that


with n,m=1,2,,2N. Then Eq. (57) can be rewritten as


or in a more compact form


The above equation givess


Note that the choice of poles, zn,n=12N, should make detI+D nonzero and I+D an invertible matrix. On the other hand, Eq. (56) can be rewritten as


Substituting Eq. (76) into (77), we thus attain


where use is made of Appendix A.1 and


In the end, by substituting (67) and (78) into (54), we attain the N-soliton solution to the DNLS+Eq. (3) under NVBC and reflectionless case (note that the time dependence of soliton solution naturally emerges in cnt):




The solution has a standard form as (80), that is


which can be proved by direct calculation for the N=1 case and by some special algebra techniques for the N>1 case.


3. Verification of standard form and the explicit breather-type multi-soliton solution

3.1 Verification of detI+B=detI+B

In order to prove the first identity in (82), we firstly calculate D¯N=detI+B. By use of (60)(62), Binet-Cauchy formula, (Appendix (A.2)) and an important determinant formula, (Appendix (A.3)), we have


here Bn1n2nr is a r,th-order principal minor of B consisting of elements belonging to not only rows n1n2nr but also columns n1n2nr. Due to (62),


in (84), n,mn1n2nr. The technique of calculating Bn1n2nr is to couple term izn2ρ4zm21 with term izm2ρ4zn2 into pair, (nm), in the denominator of n,m, (with totally rr1/2 pairs), and transplant them into the denominator of n<m, and combine with izn2zm2iρ4zm2ρ4zn2 in n<m to form a typical factor as a whole, (with just totally rr1/2 pairs). Note that if we define


and further define that


then the typical factor is




here n,mn1n2nr, and a typical function Fn is defined as


where use is made of formula (60) and (62), the time dependence of the solution naturally emerged in cnt. Substituting Eq. (88) into (83) thus completes the computation of D¯N.

Secondly, let us calculate detI+D. By use of (72) and (73), Binet-Cauchy formula, (Appendix A.2) and an important matrix formula, (Appendix A.3), we have


where Dn1n2nr is the principal minor of a r,th-order submatrix of D consisting of elements belonging to not only rows n1n2nr but also columnsn1n2nr, and


n,mn1n2nr. Using the same tricks as used in dealing with (84) leads to


Thirdly, let us calculate detI+DfTgdetI+B with BDfTg.

According to (79) and Binet-Cauchy formula (Appendix (A.2)), similarly we have


n,mn1n2nr. Using the same tricks as that used in treating (84) leads to


Due to (95), comparing (83) and (93) results in the expected identity and completes the verification of the first identity in (82).

3.2 Verification of detI+D¯=detI+B

Our most difficult and challenging task is to prove the second identity in (82). For convenience of discussion, we define that






Then the sequence of poles (25) is just in the same order as follows


On the other hand, due to (28), (62), and (73), we have




Substituting z¯n=ρ2zn̂1,z¯m=ρ2zm̂1 into above formula and using following relation


We can get an important relation between Dnm and Bm̂n̂


On the other hand, an unobvious symmetry between matrices Bnm2N×2N and Bn̂m̂2N×2N is found


It can be rewritten in a more explicit form


The last equation in (105) is due to (97) and (99), thus from (103) and (104), we have


The determinants of matrices at both sides of (106) are equal to each other


The left hand of (107) is just detI+B, and this completes verification of identity (82). From the verified (82), we know multi-soliton solution (80) is surely of a typical form as expected.

3.3 The explicit N-soliton solution to the DNLS+ equation with NVBC

In order to get an explicit N-soliton solution to the DNLS+Eq. (1) with NVBC, firstly we need to make an inverse Galileo transformation of (2) by xtxρ2tt in Fnxt in (89). Due to (51) and (85)(87), the typical soliton kernel function Fn can be rewritten as


where in (114) the n’th pole-dependent constant factor has been absorbed by redefinition of the n’th soliton center and initial phase in (110)(111).

Secondly, we need to calculate determinant CN=detI+AdetI+BfTg. According to the definition of A in (68)(69), using Binet-Cauchy formula, (Appendix (A.2)), leads to


where An1n2nr is the determinant of a r,th-order minor of A consisting of elements belonging to not only rows n1n2nr but also columnsn1n2nr.


n,mn1n2nr. Using the same tricks as used in dealing with (84) leads to


n,mn1n2nr. Substituting (108)(115) into (88) and (83) gives the explicit values of DN¯detI+B and DN. Substituting (118) into (116) then completes calculation of CN in (81). In the end, by substituting (83) and (116) into (80), we thus attain an explicit breather-type N-soliton solution of the DNLS+Eq. (1) with NVBC under reflectionless case, based upon a revised and improved inverse scattering transform. Due to the limitation of space, the asymptotic behaviors of the N-soliton solution are just similar to that of the pure N-soliton solution in Ref. [7] and thus not discussed here, but it should be emphasized that in the limit of t±, the N-soliton solution surely can be viewed as summation of N single solitons with a definite displacement and phase shift of each soliton in the whole process of elastic collisions.


4. The one and two-soliton solutions to DNLS+ equation with NVBC

We give two concrete examples – the one and two breather-type soliton solutions in illustration of the general explicit N soliton formula.

In the case of one-soliton solution, N=1,z1ρeδ1eiβ1, z2=ρ2z¯11=ρeδ1eiβ1, and δ1>0,β10π/2, using formula (82), (88), (116), (108)(115), and


we have


where not as that in (114), we define


It is different slightly from the definition in Eq. (114) for the reason that an additional minus sign “” before b20 can support (131)(133) due to b20=b¯10. Substituting (121)(122) into the following formula gives the one-soliton solution of DNLS+Eq. (1) with NVBC.


which is generally called a breather solution and shown as Figure 2.

Figure 2.

The evolution of one-breather solution in time and space.

Formula (130) includes the one-soliton solution of the DNLS equation with VBC as its limit case. In the limit of ρ0,δ1 but an invariant ρeδ1, we have


Substituting (131) and (132) into (130), we can attain


If we redefine z1ρeδ1eiπ/2β1,z2ρeδ1eiπ/2β1, then u¯1xt=q1xt, the complex conjugate of one-soliton solution (133), completely reproduce the one-soliton solution that gotten in [17, 18, 19, 20, 23], under the VBC limit with ρ0,δ1, but ρeδ1=2λ1 invariant, up to a permitted global constant phase factor. This verifies the validity of our formula of N-Soliton solution and the reliability of the newly revised inverse scattering transform.

The degenerate case for N=1, or the so-called pure one soliton solution, is also a typical illustration of the present improved IST. It can be dealt with by letting δ10. The simple poles z1=ρeiβ1 and z2=ρ2z¯11=ρeiβ1 are coincident, so do z3=z1 and z4=ρeiβ1. Meanwhile μ10,φ10, ν1=ρ2sin2β1, ib10R. Especially for the degenerate case, we have


with ν1=ρ2sin2β1, υ1=ρ21+2cos2β1, ε=sgnib10. Then we have


where ε=11 corresponds to dark (bright) soliton. Similarly if we redefine that β1π/2β1, then solution (137) is just the same as that gotten in [4, 5, 11, 12, 16] and called one-parameter pure soliton. This further convinces us of the validity and reliability of the newly revised IST for NVBC.

In the case of breather-type two-soliton solution, N=2, we define that


which is just the same as that defined in (108)(115), the pole zj-related constant complex factor is absorbed into the j’th soliton center and the initial phase. Using formula (80)(82), (88), (116), (108)(115), we have


Similarly we can attain C2 from (116) and (118) as follows


Substituting (141)(142) into (140) completes the calculation of breather-type two-soliton solution. The evolution of breather-type two-soliton solution with respect to time and space is given in Figure 3. It clearly display the whole process of the elastic collision between two breather solitons, and in the limit of infinite time t±, the breather-type two-soliton is asymptotically decomposed into two breather-type 1-solitons.

Figure 3.

Evolution of the square amplitude of a breather-type two-soliton with respect to time and space ρ=2; δ1=0.4; δ3=0.6; β1=π/5.0; β3=π/2.2; x10=0; x20=0 =0; x30=0; x40=0; φ10=0; φ20=0; φ30=0; φ40=0.

4.1 Explicit pure N-soliton solution to the DNLS+ equation with NVBC

When all the simple poles are on the circle Oρ centered at the origin O, just as shown in Figure 4, our revised IST for DNLS+ equation with NVBC will give a typical pure N-soliton solution. The discrete part of az is of a slightly different form from that of the case for breather-type solution, and it can be expressed as

Figure 4.

Integral contour as all poles are on the circle of radiusρ.


At the zeros of az, we have


On the other hand, the zeros of az appear in pairs and can be designed by zn, (n=1,2,,N), in the I quadrant, and zn+N=zn in the III quadrant. The Zakharov-Shabat equation for pure soliton case of DNLS+ equation under reflectionless case can be derived immediately


Here Λ=κλ,Λn=κnλn; Letting z=ρ2zm1, m=1,2,,N, then


Different from that in breather-type case, we define znρeδneiβn=ρeiβn, with βn0π/2,δn=0, i=12N, specially we have


An inverse Galileo transformation xtxρ2tt changes Fnxt into


Due to b¯n0=bn0,ibn0R, following equations hold:




where En is also a real constant which is only dependent upon the order number n. The constant and positive real number ibn0 has been absorbed by redefinition of the n’ th soliton center xn0 in (155). Thus the determinants in formula (83) for pure soliton solution can be calculated as follows


Substituting (149)(157) into (158)(161), and substituting (158)(161) into the following formula, we attain the explicit pure N-soliton solution


The N=2 case, that is, the pure two-soliton is also a typical illustration of the general explicit N-soliton formula. According to (158)(162), it can be calculated as follows


The evolution of pure two-soliton solution with respect to time and space is given in Figure 5. It clearly demonstrates the whole process of the elastic collision between pure two solitons. If 0<β2<β1<π/2, then ε1=1,sgnE1=1 and ε2=1,sgnE2=1 correspond to double-dark pure 2-soliton solution as in Figure 5a; ε1=1,sgnE1=1 and ε2=1,sgnE2=1 correspond to a double-bright pure 2-soliton solution in Figure 5c; ε1=1,sgnE1=1 and ε2=1,sgnE2=1 correspond to a dark-bright-mixed pure 2-soliton solution in Figure 5b. In the limit of infinite time t±, the pure 2-soliton solution is asymptotically decomposed into two pure 1-solitons.

Figure 5.

Evolution of pure two soliton solution in time and space. (a) dark-dark pure 2-soilton, (b) dark-bright pure 2-soilton, and (c) bright-bright pure 2-soilton.

By the way, it should be point out, although our method and solution have different forms from that of Refs. [7, 16], they are actually equivalent to each other. In fact if the constant En, (n=1,2,,N), is also absorbed into the n’th soliton center xn0 just like ibn0 does in (152)(154), and replace βn with βn=π/2βn0π/2, the result for the pure soliton case in this section will reproduce the solution gotten in Refs. [7, 9, 16].

On the other hand, letting only part of the poles converge in pairs on the circle in Figure 1 and rewriting the expression of anz as in Ref. [7, 8, 12], our result can naturally generate the mixed case with both pure and breather-type multi-soliton solution.

4.2 The asymptotic behaviors of the N-soliton solution

Without loss of generality, we assume β1>β2>>βn>>βN; υ1<υ2<<υn<<υN in (156), and define the n’th neighboring area as ϒn:xxnoυnt0,n=1.2N. In the neighboring area of ϒn,




In the neighboring area of ϒn, we have




As t, the N neighboring areas queue up in a descending series ϒN,ϒN1,,ϒ1, then


the N-soliton solution can be viewed as N well-separated exact pure one solitons, each u1θn+Δθn, (1,2,,n) is a single pure soliton characterized by one parameter βn, moving to the positive direction of the x-axis, queuing up in a series with descending order number n.

As t, in the neighboring area of ϒn we have




That is, the N-soliton solution can be viewed as N well-separated exact pure one solitons, queuing up in a series with ascending order number n such as ϒ1,ϒ2,,ϒN..

In the process of going from t to t, the n’th pure single soliton overtakes the solitons from the 1’th to n1’th and is overtaken by the solitons from n+1’th to N’th. In the meantime, due to collisions, the n’th soliton got a total forward shift Δθn/νn from exceeding those slower soliton from the 1’th to n1’th, got a total backward shift Δθn+/νn from being exceeded by those faster solitons from n+1’th to N’th, and just equals to the summation of shifts due to each collision between two solitons, that is,


By introducing an suitable affine parameter in the IST and based upon a newly revised and improved inverse scattering transform and the Z-S equation for the DNLS+ equation with NVBC and normal dispersion, the rigorously proved breather-type N-soliton solution to the DNLS+ equation with NVBC has been derived by use of some special linear algebra techniques. The one- and two-soliton solutions have been given as two typical examples in illustration of the unified formula of the N-soliton solution and the general computation procedures. It can perfectly reproduce the well-established conclusions for the special limit case. On the other hand, letting part/all of the poles converge in pairs on the circle in Figure 4 and rewriting the expression of anz as in [7, 12, 13], can naturally generate the partly/wholly pure multi-soliton solution. Moreover, the exact breather-type multi-soliton solution to the DNLS+equation can be converted to that of the MNLS equation by a gauge-like transformation [17].

Finally, the elastic collision among the breathers of the above multi-soliton solution has been demonstrated by the case of a breather-type 2-soliton solution. The newly revised IST for DNLS+ equation with NVBC and normal dispersion makes corresponding Jost functions be of regular properties and asymptotic behaviors, and thus supplies substantial foundation for its direct perturbation theory.


5. Space periodic solutions and rogue wave solution of DNLS equation

DNLS equation is one of the most important nonlinear integrable equations in mathematical physics, which can describe many physical phenomena in different application fields, especially in space plasma physics and nonlinear optics [1, 2, 16, 24, 25, 26, 27, 28, 29]. We have found that DNLS equation can generate not only some usual soliton solutions such as dark/bright solitons and pure/breather-type solitons, but also some special solutions – space periodic solutions and rogue wave solution [14].

There are two celebrated models of the DNLS equations. One equation is called Kaup-Newell (KN) equation [15]:


and the other is called Chen-Lee-Liu (CLL) equation [30]:


Actually, there is a gauge transformation between these two Eqs. (183) and (184) [14, 30, 31]. Supposing u is one of the solutions of the KN Eq. (183), then


will be the solution of the CLL equation.

This section focuses on the KN Eq. (183) with NVBC – periodic plane-wave background. The first soliton solution of (183) was derived by Kaup and Newell via inverse scattering transformation (IST) [3, 15, 32]. Whereafter, the multi-soliton solution was gotten by Nakamura and Chen by virtue of the Hirota method [30, 31]. The determinant expression of the N-soliton solution was found by Huang and Chen on the basis of the Darboux transformation (DT for brevity) [33], and by Zhou et al., by use of a newly revised IST [7, 11, 12, 13, 17].

Recently, rogue waves which seem to appear from nowhere and disappear without a trace have drawn much attention [34, 35]. The most significant feature of rogue wave is its extremely large wave amplitude and space-time locality [35]. The simplest way to derive the lowest order of rogue wave, that is, the Peregrine solution [35, 36], is to take the long-wave limit of an Akhmediev breather [37] or a Ma breather [38], both of which are special cases of the periodic solution. Thus, the key procedure of generating a rogue wave is to obtain an Akhmediev breather or a Ma breather. As far as we know, DT plays an irreplaceable role in deriving the rogue wave solution [39, 40, 41]. Because both Akhmediev breather and Ma breather can exist only on a plane-wave background; Darboux transformation has the special privilege that a specific background or, in other words, a specific boundary condition can be chosen as the seed solution used in DT. For instance, if we choose q0=0 as the seed solution of the DT of the KN Eq. (183), then after 2-fold DT, a new solution will be gotten under VBC:


(where all the parameters are defined in Ref. [38]). Similarly, setting a seed solution q0=cexpiax+c2+aat, a plane-wave solution to Eq. (183), will generate a new solution after 2-fold DT under a plane-wave background. Therefore, there is no need to discuss the boundary conditions or background when applying DT to solve those nonlinear integrable equations. This makes DT the most effective and prevailing method in obtaining a rogue wave solution.

Compared with DT, the IST has its fatal flaw that the difficulty of dealing with the boundary condition is unavoidable, which limits the possible application of the IST. Although the KN equation has been solved theoretically by means of an improved IST for both VBC and the NVBC [7, 8, 9, 17, 18, 19, 20], there is no report that the KN equation could be solved under a plane-wave background by means of IST. And consequently, it appears that rogue wave solutions cannot be obtained through the IST method. This major problem is caused by the difficulty of finding appropriate Jost solutions under the plane-wave background.

On the other hand, the Hirota’s bilinear-derivative transform (HBDT for brevity) [42, 43, 44, 45, 46], though not as a prevalent method as DT, has its particular advantages. The core of this method is a bilinear operator D which is defined by:


where, at the left side of the above formula, a dot between two functions Axt and Bxt represents an ordered product. The HBDT method is very useful in dealing with periodic solutions for its convenience in computing the bilinear derivatives of an exponential function [44]:


Here, F represents general function expressed by the finite or infinite power series expansion of the Hirota’s bilinear differential operators. Formula (188) is the generalization of Appendix 5.5. Thus using HBDT method to find space periodic solutions of KN equation is practicable. The space periodic solutions possess the characters that they approach the plane-wave solution when t and are periodic in space. The first space periodic solution was found by Akhmediev with one parameter [45]. Actually, we can regard the space periodic solution as a special Akhmediev breather with a pure complex-valued wave number. Further, through a space periodic solution, a rogue wave solution can be constructed. This means besides DT, HBDT method is also an alternative and effective way to find rogue wave solution of KN equation.

5.1 Bilinear derivative transformation of DNLS equation

The Hirota bilinear transformation is an effective method which could help to solve the KN equation. Due to the similarity of the first equation of Lax pairs between that of DNLS equation and AKNS system, there is a direct inference and manifestation that uxt has a typical standard form [6, 7]:


where f and g are complex auxiliary functions needed to be determined. Applying the bilinear derivative transform to (189), we can rewrite the derivatives of uxt in the bilinear form [19, 20, 42, 43, 44, 45, 46]:


Directly substituting the above Eqs. (190)(192) into (183) gives:


Then the above transformed KN equation can be decomposed into the following bilinear equations:


where λ is a constant which needs to be determined. Notice that if λ=0 then the above bilinear equations are overdetermined because we have only two variables but three equations. Actually, setting λ=0 is the approach to search for the soliton solution of the DNLS equation under vanishing boundary condition [19, 20]. Here, we set λ as a nonzero constant to find solutions under a different boundary condition – a plane-wave background.

5.2 Solution of bilinear equations

5.2.1 First order space periodic solution and rogue wave solution

Let us assume that the series expansion of the complex functions f and g in (189) are cut off, up to the 2’th power order of ϵ, and have the following formal form:


Substituting f and g into Eqs. (194)(196) yields a system of equations at the ascending power orders of ϵ, which allows for determination of its coefficients [14, 19, 20]. We have 15 equations [14, 19, 20] corresponding to the different orders of ϵ. After solving all the equations, then we can obtain the solution of the DNLS equation:






Notice that ρ and M are real; b1 and φ0 are complex constants, so there are two restrictions for a valid calculation: (1) the wave number p must be a pure imaginary number; (2) the angular frequency Ω must not be purely imaginary number and must furthermore satisfy the quadratic dispersion relation:


According to the test rule for a one-variable quadratic, there is a threshold condition under which Ω will not be a pure imaginary number:


The asymptotic behavior of this breather is apparent. Because the wave number p is a pure imaginary number, the breather is a periodic function of x. The quadratic dispersion relation (204) permits the angular frequency Ω to have two solutions:


If we set Ω=Ω+, because 2p4+p2ρ4>0, then t will lead to:


And t will lead to:


where φ is the phase shift across the breather:


and due to a1a2=b1b2, thus the above phase shift φ is real and does not affect the module of the breather u1 when t. As for the other choice Ω=Ω, further algebra computation shows the antithetical asymptotic behavior of g1, f1, and u1 when t. In a nutshell, u1 will degenerate into a plane wave.

Hereto, we have completed the computation of the 1st-order space periodic solution, the space-time evolution of its module is depicted in Figure 6. In what follows, we will take the long-wave limit, that is, p → 0, to construct a rogue wave solution. Supposing p=iq, here q is a real value and q0, then the asymptotic expansion of the angular frequency Ω is:

Figure 6.

The space-time evolution of the module of the 1st order space periodic solution in (198) with p=i,ρ=2,b1=i and Ω=Ω+, complex constant φ0 is set to zero.


where σ=±2. For the sake of a valid form of the rogue wave solution, we need to set b1=1 and φ0=0 (of course, setting b1=1 and eφ0=1 is alright, all we need is to make sure that the coefficients of the q0 and q1 in the expansions of f1 and g1 are annihilated). Therefore, the expansions of g1 and f1 in terms of q are given by:


Consequently, the rogue wave solution can be derived according to Eq. (198):




Here ω and β are given by Eq. (201), ρ is an arbitrary real constant. The module of rogue wave solution Eq. (218) is shown in Figure 7.

Figure 7.

The space-time evolution of the module of the rogue wave solution with ρ=1 and σ=2. The max amplitude is equal to 3 at the point x=2t=22/3.

As we discussed in the Introduction section, there is a gauge transformation between KN Eq. (183) and CLL Eq. (184). Thus, it is instructive to use the integral transformation Eq. (185) to construct a solution of Eq. (184). Substituting the solution (198) into (185), further algebra computation will lead to a space periodic solution of the CLL equation:


where, g1, f1, and other auxiliary parameters are invariant and given by Eqs. (199)(203). The same procedures which are used to derive the rogue wave solution of the KN equation can be used to turn υc into a rogue wave solution of the CLL equation:


which has the same parameters as uRW. And this solution υc,RW has exactly the same form as the result given by ref. [46].

5.2.2 Second-order periodic solution

Taking the similar procedures described previously could help us to derive the 2nd-order space periodic solution. Assume the auxiliary functions f and g to have higher order expansions in terms of ϵ:


Similarly, substituting f and g into the bilinear Eqs. (194)(196) leads to the 27 equations [14, 19, 20] corresponding to different orders of ϵ. Solving these equations is tedious and troublesome but worthy and fruitful. The results are expressed in the following form:




where i,j,k=1,2,3,4, and the above parameters and coefficients are given respectively by:


Of course, for a valid and complete calculation, we are faced with the same situation as the 1st-order breather: ρ is real, b1,b3 and all φ0i are complex constants. Certainly, each wave number pi must be a pure imaginary number and each angular frequency Ωi has to satisfy the quadratic dispersion relation:

Figure 8.

The space-time evolution of the module of the 2nd order space periodic solution with p1=0.4i,p3=0.75i,b1=i,b3=1 and ρ=1.6. Other phase factors φ1 and φ3 are set to zero.


And the threshold conditions for each complex-valued Ωi share the same form as Eq. (205):


The space-time evolution of the module of the 2nd order space periodic solution (223) is shown in Figure 8. Paying attention to the form of this breather and the previous one, we will notice that this breather can exactly degenerate into the 1st-order breather if we take p3=p1. Under this condition, M13=M24=0, thus the higher order interaction coefficients Tijk and A will vanish. Therefore, g2 and f2 will degenerate into the forms of g1 and f1, respectively:


where b1=χb1,b2=χ¯b2,a1=χa1 and a2=χ¯a2 with χ=b1+b3/b1. That is how u2 can be reduced to u1. Given to this reduction, a generalized form of these two breathers arises:


where the coefficient M is defined by:


On the other hand, this breather possesses the same feature as the former one that it is periodic with respect to variable x due to the pure imaginary numbers p1 and p3. In addition, its asymptotic behaviors are analogical to the 1st-order space periodic solution. Each quadratic dispersion equation has two roots, respectively:


Thus, we will have four combinations of Ω1 and Ω2. Details are numerated in Table 1. The parameters φ0,φ and φ in Table 1 are the phase shifts which are all real so that they will not change the module of u2 when t. And φ is given in Eq. (214), and others are determined by:

Choice of ΩiΩ1+,Ω2+Ω1+,Ω2Ω1,Ω2+Ω1,Ω2

Table 1.

Asymptotic behaviors of u2.


From Table 1, we could draw the conclusion that this breather will also degenerate into the background plane wave as t. Furthermore, there is a phase shift across the breather from t= to t=, which depended on the choice of Ω1 and Ω2.

In this section, the 1st order and the 2nd order space periodic solutions of KN equation have been derived by means of HBDT. And after an integral transformation, these two breathers can be transferred into the solutions of CLL equation. Meanwhile, based on the long-wave limit, the simplest rogue wave model has been obtained according to the 1st order space periodic solution. Furthermore, the asymptotic behaviors of these breathers have been discussed in detail. As |t| → ∞, both breathers will regress into the plane wave with a phase shift.

In addition, the generalized form of these two breathers is obtained, which gives us an instinctive speculation that higher order space periodic solutions may hold this generalized form, but a precise demonstration is needed. Moreover, higher order rogue wave models cannot be constructed directly by the long-wave limit of a higher order space periodic solution because the higher order space periodic solution has multiple wave numbers pi, we are also interested in seeking an alternative method besides DT that could help us to determine the higher order rogue wave solutions.


6. Concluding remarks

In the end, as the author of the above two parts, part 1 and 2, I want to give some concluding remarks. As a whole, the two parts had taken the DNLS equation as a reference, systematically introduced several principal methods, such as IST, GLM (Marchenko) method, HBDT, to solve an integrable nonlinear equation under VBC and NVBC. We had gotten different kinds of soliton solutions, such as the light/dark soliton, the breather-type soliton, the pure soliton, the mixed breather-type and pure soliton, and especially the rogue-wave solution. We had also gotten soliton solutions in a different numbers, such as the one-soliton solution, the two-soliton solution, and the N-soliton solution. Nevertheless, I regret most that I had not introduced the Bäcklund transform or Darboux transform to search for a rogue wave solution or a soliton solution to the DNLS equation, just like professor Huang N.N., one of my guiders in my academic research career, had done in his paper [33]. Another regretful thing is that, limited to the size of this chapter, I had not introduced an important part of soliton studies, the perturbation theory for the nearly-integrable perturbed DNLS equation. Meanwhile, this chapter have not yet involved in the cutting-edge research of the higher-order soliton and rogue wave solution to the DNLS equation, which remain to be studied and concluded in the future.


Some useful formulae.

A1, If A1 and A2 are N×1 matrices, A is a regular N×N matrix, then


A2, Binet-Cauchy formula: For a squared N×N matrix B


whereBn1n2nr is a r’th-order principal minor of B.

A3, For a N × N matrix Q1 and a N × N matrix Q2,



where Q1n1n2nrm1m2mr denotes a minor, which is the determinant of a submatrix of Q1 consisting of elements belonging to not only rows (n1,n2,nr) but also columns (m1,m2,,mr).

The above formula also holds for the case of detI+Ω1Ω2 With Ω1 to be a N×N+1 matrix and Ω2 a N+1×N matrix.

A4, For a squared matrix C with elements Cjk=fjgkxjyk1,


A5, Some useful blinear derivative formulae.


where ηi=Ωit+Λix+η0ii=1,2; Ωi, Λi, η0i are complex constants.


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Written By

Zhou Guo-Quan

Submitted: July 2nd, 2020 Reviewed: July 22nd, 2020 Published: September 22nd, 2020