Darboux Transformations,Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schr dinger Equation∗

Mi Chen(陈觅),Biao Li(李彪), and Ya-Xuan Yu(于亚璇)

Department of Mathematics,and Ningbo Collaborative Innovation Center of Nonlinear Harzard System of Ocean and Atmosphere,Ningbo University,Ningbo 315211,China

Abstract By Taylor expansion of Darboux matrix,a new generalized Darboux transformations(DTs)for a(2+1)-dimensional nonlinear Schr dinger(NLS)equation is derived,which can be reduced to two(1+1)-dimensional equation:a modified KdV equation and an NLS equation.With the help of symbolic computation,some higher-order rational solutions and rogue wave(RW)solutions are constructed by its(1,N−1)-fold DTs according to determinants.From the dynamic behavior of these rogue waves discussed under some selected parameters,we find that the RWs and solitons are demonstrated some interesting structures including the triangle,pentagon,heptagon profiles,etc.Furthermore,we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter.These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.

Key words:Darboux transformations,nonlinear Schr dinger equation,higher-order rational solution,rogue wave solution

1 Introduction

There are many methods to obtain some exact solutions to the integrable nonlinear systems,for example,Darboux transformation(DT),inverse scattering transformation,Hirota bilinear method,and so on.[1−3]Among them,DT is usually a more valid and powerful method to investigate multi-soliton solutions of an integrable system from a simple seed solution.In 2009,the DT is modified to obtain the higher-order Rogue waves(RW)solutions of the focusing NLS equation,[4]but it cannot give an accurate formulae for multi-RW solutions.Recently,the Matveev’s generalized DT method[5]was developed to construct the higher-order RW solutions of the NLS and KN equations.[6−7]More Recently,Wen et al.presented a novel and simple method to find the generalized(n,M)-fold DT of the(2+1)-dimensional KP equation and the(2+1)-dimensional generalized KP equation and obtained their higher-order rogue waves and rational solutions.[8−9]

where u=u(x,y,t)and V=V(x,y,t),which is derived by dimensionally reducing the self-dual Yang Mills equation by Strachan.We know that the(2+1)-dimensional NLS equation is of great importance in fluid mechanics and nonlinear theory.[26−28]From a moving space curve formalism,geometrical as well as gauge equivalence between a(2+1)-dimensional spin equation(M-I equation)and the(2+1)-dimensional NLS originally have been established.[29−31]Its geometrical properties have been studied in Ref.[32].Radha et al.investigated its Painlev´e property and obtained its exact solutions by using Hirota’s bilinear method.[33]One and two-soliton solution of Eq.(1)are obtained in Ref.[34]and dromionic ones can be found in Ref.[35].Our work is mainly to study its generalized N-order DTs in terms of the Taylor series expansion and a limit procedure to directly obtain the higher-order rational solutions and RW solutions of Eq.(1).From the process of this method,we can see that some high-order RW solutions can be obtained by relatively simple iterations,and the relationship between high-order RW solutions and the seed solution can be derived simply.

The outline of this paper is as follows.In Sec.2,we firstly review the usual DT of Eq.(1),then following the idea of Refs.[8–9],we derive the new,generalized perturbation(n,M)-fold DT for the(2+1)-dimensional NLS equation by using the Taylor expansion and a limit procedure.In Sec.3,by selecting generalized perturbation(1,N−1)-fold DTs,we obtain its higher-order rational solutions and RW solutions according to determinants with help of symbolic computation and discuss their rich wave structures.Some conclusions and discussions are given in the last section.

2 Rational Solutions and Rouge Waves of the(2+1)-Dimensional NLS Equation

The(2+1)-dimensional NLS equation can be separated into two(1+1)-dimensional equations[36]as follows:

If u=ψ is fit for NLS equation

then Eq.(1)is reduced to(1+1)-dimensional complex mKdV equation

In line with the decomposition,a unified new DT will be constructed and more solutions with several free parameters of Eq.(1)will be found.Next,we will find the solution of Eq.(1)by solving the solution ψ(x,y,t)of Eqs.(2)and(3).

The Lax pair(i.e.,linear spectral problems)of Eqs.(2)and(3)is

with

where the star denotes the complex conjugation,the vector Φ =(ϕ,φ)Tis an eigenfunction associated with spectrl parameter λ,and i2= −1.It is easy to verify that Eqs.(2)and(3)can be generated by two zero curvature equations My−Vx+[M,V]=0 and Mt−Wx+[M,W]=0.

2.1 Lax Pair and Darboux Transformation

We introduce the gauge transformation with the Darboux matrix T(λ):

and we can obtain

where the generalized bracket for the square matrix is defined byBased on Ref.[37],the usual N-order Darboux matrix T(λ)is chosen as follows

where N is a positive integer,Ai,Bi(0 6 i 6 N−1)are 2N unknown complex functions and satisfy the following 2N equations

where Φs(λs)=(ϕs(λs), φs(λs))T,(s=1,2,...,N)are the solutions of Lax pair(4)for the spectral parameters λsand the initial solution ψ0.

It is easy to know that the system(12)has 2N distinct parameters λk,(λi̸= λj,λi̸=0,i ̸=j,i=1,2,...,N)which are the roots of the 2N order polynomial detT(λ),i.e.,

The following Darboux transformation can be found in Ref.[37],then Eq.(8)holds the Darboux matrix(11).

Theorem 1Let Φi(λi)=(ϕi(λi), φi(λi))T(i=1,2,...,N)be N distinct column vector solutions of the spectral problem(4)for the corresponding N distinct spectral parameters λ1, λ2,...,λNand the initial solution ψ0of Eqs.(2)and(3),respectively,then the N-fold Darboux transformation of Eq.(1)is as follows

where B(N−1)= △B(N−1)/△N,with

where∆B(N−1)is given by the determinant∆Nby replacing its(N+1)the column with the column vector

2.2 Generalized Perturbation(1,N−1)-fold Darboux Transformation

Next we will discuss generalized perturbation(1,N−1)-fold Darboux transformation.We can select some functions Aiand Biin the above-mentioned Darboux matrix T and the initial solution ψ0to obtain some solutions of Eq.(1)such as multi-rogue wave solutions.We maintain spectral parameter λ = λ1not N spectral parameters λ=λk(k=1,2,...,N)of the Darboux matrix(11),from the condition T(λ1)Φ(λ1)=0 we can derive the following system

where(ϕ(λ1),φ(λ1))Tis a solution of the linear spectral problem(8).Next we can expand the expression T(λλ1)Φ(λ1)|λ1=λ1+ϵ=T(λ1+ϵ)Φ(λ1+ϵ)at ϵ=0.As we know that

with Φ(k)(λ1)=(1/k!)(∂k/∂λk)Φ(λ)|λ=λ1and

where T(0)(λ1)=T(λ1),T(N)(λ1)=I,and

Let

we can determine the 2N unknown functions Ai,Bi,(0 6 i 6 N−1)in Eq.(11).So based on Eq.(19),we can obtain the linear algebraic system with the 2N equations,

We can choose a suitable eigenvalue λ1in order to the determinant of the system(20)is non-zero.So the transformation matrix T can be uniquely determined by the system(20).Then we can obtain the new DT with the same eigenvalue λ = λ1by new distinct functions Ai,Biobtained in the N-order Darboux matrix T.

Theorem 2Let Φ(λ1)=(ϕ(λ1),φ(λ1))Tbe a distinct column vector solutions of the spectral problem(4)for the corresponding spectral parameters λ1and the initial solution ψ0of Eqs.(2)and(3),respectively,then the(1,N−1)-fold Darboux transformation of Eq.(1)is as follows

where B(N−1)= △B(N−1)/△N,with

here∆i,k(1 6 i,k 6 2N),with

and(N+1)-th column with the vector b=(bi)2N×1of∆Ncan produce∆B(N−1),with

2.3 Generalized Perturbation(n,M)-fold Darboux Transformation

Now we investigate the Darboux matrix(11)and assume that the eigenfunctions Φi(λi)(i=1,2,...,n)are the solutions of Eq.(8)with λiand initial solution ψ0of Eqs.(2)and(3).So we obtain

with i=1,2,...,n and ki=0,1,...,mi.According to the idea of Section 2.2,we can get the 2N equations

Comparing system(22)with system(12),it is easy to see that the first several systems for every index i are the same,when there exist at least one index mi̸=0 then they are different.

Theorem 3Let Φi(λi)=(ϕi(λi),φi(λi))T(i=1,2,...,n)be a distinct column vector solutions of the spectral problem(4)for the corresponding the spectral parameters λi(i=1,2,...,n)and the initial solution ψ0of Eqs.(2)and(3),respectively,the(n,M)-fold Darboux transformation of Eq.(1)is as follows

and∆B(N−1)is obtained from the determinantby replacing its(N+1)-th column with the vector(b(1)···b(n))T,where

By solving Eq.(22),we can obtain the new functions Aiand Biin the N-order Darboux matrix different from in the usual DT transformation.So we can derive some new solutions by the new(1,N−1)-fold DT with the same eigenvalue λ = λ1or the(n,M)-fold DT with n distinct eigenvalues λi(i=1,2,...,n).We call Eqs.(5)and(23)as a generalized perturbation(n,M)-fold DT of Eq.(1).

3 The Higher-Order Rational Solutions and Rouge Waves Solutions of the(2+1)-Dimensional NLS Equation

In this section,we will use the generalized perturbation(1,N−1)-fold DT to present some higher-order rational solutions and RW solutions of Eq.(1).Differing from the chosen zero seed solution to study multi-soliton solutions,[18]here we select a plane wave solution as a seed solution of Eqs.(2)and(3)as the following

where a is a real parameter,the wave numbers in x-and y-directions are a and a2−2,respectively.From Eq.(24)it is easy to derive that the phase velocities in x-and y-directions are(6−a2)and a(6−a2)/(a2−2),respectively,and the group velocities in x-and y-directions are 3(2−a2)and 3−2a2,respectively.

The eigenfunctions solution of the Lax pair(4)with Eq.(24)is as follows:

with

where pi,qi(i=1,2,...,N)are all real parameters and ϵ is a small parameter.

Next,by setting the eigenvalue λ =a+2i+ ϵ2and expanding the vector function Φ(ϵ2)in Eq.(25)as a Taylor series at ϵ=0,[8]we derive to the following results

(ϕ(i),φ(i))T(i=2,3,...)are cumbersome,so we omit them here.

Next we discuss the solution(21)from the following four cases(N=1,2,3,4)to comprehend the obtained exact solutions of Eq.(1).According to solution(21),for n=1,we can only give the trivial constant solution.In this case,we do not use the derivatives of T(λ1)Φ(λ1)to determine B(0)so we cannot obtain a new solution.

Case 1For N=2,from the generalized perturbation(1,1)-fold DT,we can obtain the first-order RW solution(regular rational solution)of the(2+1)-dimensional NLS Eq.(1).

with

where a is a free parameter.

Based on Eq.(26)we can obtain the maximum √and minimum values of the modulus ofWhen x=3t(a2+2)and y=−3at,we have the maximum valueand when x=3t(a2+2)± and y=−3at,we have the minimum valueHere we consider t as arbitrary real number.When x,t→∞,

It needs to be pointed out that when a=0,Eq.(1)has the solution as the following

when y=0,u(x,y,t)is the solution of the(1+1)-dimensional complex mKdV equation.That is to say,we can use the reduction y=0 of the RW solution of the(2+1)-dimensional NLS equation to obtain the RW solution of the(1+1)-dimensional complex mKdV equation.It also fits for the following higher-order RW solutions.

Fig.1 (Color online)u1=|˜u1|2is the square of the modulus of the first-order rational solution and RW solution given by Eq.(26)at different two-dimensional spaces.(a)a=y=0,(b)a=1,y=0,(c)a=t=0,(d)a=0,x=0,(e)a=0,t=2,(f)a=0,t=3.

Keep y=0,when the parameter a=0 and a̸=0(e.g.,a=1)the wave profiles of solution(26)in(x,t)-space are obviously different.The solution(26)displays the W-shaped solitary wave with a=0,and it is not localized(see Fig.1(a)),while the solution(26)displays the first-order RW profile(see Fig.1(b)).So that when y=0,the parameter a can change the solution(26)in the(x,t)-space from the non-localized solution(a=0)to the localized solutions(a=1).From Fig.1(c)and Fig.1(d),we can know that for any parameter a,whether t=0 or x=0,the solution(37)generates the same first-order RW profiles.

Next,we mainly discuss the localized wave structure of Eq.(26)in(x,y)-space under fixed time:

We make t change and keep the parameter a=0,the solution(26)keeps the shape of the first-order RW profile;From Figs.1(a),1(b),1(c)we can see a phenomenon that the core of the first-order RW is located at the origin with t=0;The core moves positive along the x-axis,when t increases;

Fig.2 (Color online)u1= and is the first-order rational solution and RW solution given by Eq.(26).(a)a=1,t=−3,(b)a=1,t=0,(c)a=1,t=3,(d)a=−1,t=−3,(e)a=−1,t=0,(f)a=−1,t=3.

As we can see from Figs.2(a),2(b),2(c),when a̸=0,the shape of the first-order RW does not change and the graphics core remains at the origin with t=0.When time t increases or decreases,the first-order RW moves to the low right or upper left on the(x,y)-plane with a>0,and the first-order RW moves to the upper right or low left on the(x,y)-plane with a<0(see Figs.2(d),2(e),2(f)).

Case 2For N=3,on the basis of the generalized(1,2)-fold DT,the second-order RW solution of the(2+1)-dimensional NLS Eq.(1)can be derived as follows

which includes three parameters a,b1,and c1.Due to the complexity of the equation,we omitted it here.Because b1and c1have the similar effect on the solution(28),we only consider a and b1and let c1=0 with y=0.Under different parameters,we can get four cases as follows:

From Fig.3(a),we can derive that the solution(28)displays the elastic interaction of two soliton solutions at a=b1=0 and is not localized.While a̸=0(e.g.,a=1),b1=0,the solution(28)yields the strong interaction of two first-order RWs with(e.g.,a=1),b1=0(see Fig.3(b)).

From Figs.3(a),3(b),we can clearly know that the solution(28)in the(x,t)-space can be modulated by the parameter a from the non-localized solution into the localized second-order RW solutions.

When a=0 and b1̸=0(e.g.b1=100),the solution(28)is not localized and it is divided into two soliton solutions without any interaction.The amplitude of one soliton becomes high and another one becomes low as|x|,|t|increase(see Fig.3(c)).Moreover,it follows from Fig.3(c)that the width of the upper solitary wave becomes narrow and another one becomes wide as|x|,|t|increase.

When a̸=0(e.g.a=1)and b1=100,the solution(28)expresses three first-order RWs without any interaction(Fig.3(d)),so we can see the solution(28)in the(x,t)-space can be modulated from the non-localized solution into localized solutions by the parameter a.

Based on Eq.(28)we can obtain the maximum and minimum values of the modulus ofat a=0,b1=0,and c1=0.When x=0 and y=0,we have the maximum value=5,and when x=∞and y=∞,we have the minimum value

Fig.3 (Color online)u1= and is the second-order rational solution and RW solution of Eq.(28).(a)a=0,b1=0,c1=0,(b)a=1,b1=0,c1=0,(c)a=0,c1=0,b1=100,(d)a=1,b1=100,c1=0.

Fig.4 (Color online)u1= and is the second-order rational solution and RW solution of Eq.(28)(a)b1=0,t=0,(b)b1=0,t=1,(c)b1=100,t=0,(d)b1=100,t=1.

In other words,compare Figs.3(a)and 3(b),we know that the solution(28)exhibits both non-localized wave profile in(x,t)-space at a=0 with the different parameter b1.While the solution(28)represents two parallel solutions with b1=100 and it displays the strong interaction of two solutions with b1=0.Then,compare Figs.3(b)and 3(c),the solution(28)expresses both localized wave profiles in(x,t)-space with a=1.

The solution(28)performed the similar second-order RW profiles in both(x,y)-space at t=0 and(y,t)-space at x=0 with any parameters a and b1.In the following we fixed a=0,c1=0,t=0,and set different parameters b1to discuss the wave profiles of the second-order RW solution.

We have the profiles of weak interactions of second-order RWs at t=0,b1=10,(Fig.4(c))and t=1,b1=100,(Fig.4(d))with a=c1=0,and the second-order RW is split into three first-order RWs that array an isosceles triangle structure.

From(Fig.4(c))and(Fig.4(d)),we can know that as time increases,the center of triangle structure is farther away from the origin with b1(e.g.b1=100).

From(Fig.4(a))and(Fig.4(b)),we can know that the second-order RW can be converted into the separable three first-order RWs by adjusting b1and t.

Fig.5 (Color online)u1= and is the third-order RW solution of Eq.(29)(a)b1=0,b2=0,c2=0,t=0,(b)b1=0,b2=0,c2=0,t=1,(c)b1=50,b2=0,c2=0,t=0,(d)b1=50,b2=0,c2=0,t=1,(e)b1=0,b2=50,c2=0,t=0,(f)b1=0,b2=50,c2=0,t=1,(g)b1=0,b2=500,c2=500,t=0,(h)b1=0,b2=500,c2=500,t=1.

Case 3For N=4,the second-order RW solution of the(2+1)-dimensional NLS Eq.(1)can be derived by the generalized perturbation(1,3)-fold DT

Because of its complexity,we omit it here,but we can know it has five parameters(a,b1,b2,c1,c2).Next we discuss some meaningful structure of the third-order RW solution(29).

The wave structures of third-order RW are shown in Figs.5(a)–5(b)with parameters a=b1=b2=c1=c2=0 at different time.

From Figs.5(c),5(d),we can see the third-order RW is made up of the six first-order RWs,and the distribution form is a regular triangle shape with a=b2=c1,2=t=0 and b1=0,or b1>0(e.g.b1=50).

From Figs.5(e),5(f)we can see the third-order RW is made up of the six first-order RWs and the shape is a regular pentagon,while as the time increases the shape develops towards the regular triangle with b1=0,a=0,b2>0,(e.g.,b2=50),c1,2=t=0,and b1=0,a=0,b2>0,c1,2=0,t0,(e.g.t=1,b2=50).

From Figs.5(g),5(h)we can see the third-order RW is made up of the six first-order RWs and the shape is a regular pentagon with b1=0,a=0,b2>0,c1=0,c2>0,t=0,(e.g.b2=c2=500),while the six first-order RWs array an irregular pentagon with t̸0(e.g.t=1).

It is necessary to point out that for N>4,we can also find higher-order rational solutions and RW solutions of Eq.(1)with rich structures.

4 Summary and Discussion

In summary,we have constructed the new generalized perturbation(n,M)-fold Darboux transformation(DT)to find the higher-order rational solutions and rogue wave(RW)solutions of the(2+1)-dimensional NLS equation.The process is mainly divided into two steps:Firstly,a brief introduction to the usual N-fold DT for Eq.(1)is given.Secondly,the N-order Darboux matrix,the Taylor expansion and a limit procedure are used to construct the generalized perturbation(n,M)-fold DTs for Eq.(1).Next,the generalized(1,N−1)-fold DT with only one spectral parameter is chosen to obtain higher-order RW solutions of Eq.(1).RW solutions’propagation and interaction are discussed and demonstrated by some figures,which display rich and interesting wave structures including the triangle,pentagon,heptagon profiles,etc.The results of this article are general and interesting.The used idea can be also applied to other physically nonlinear wave models.Because more higher-order rational solutions contain more parameters,general spatial-temporal structures of those RWs may be expected and need further investigate in the future.