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        -dressing method for the complex modified KdV equation

        2023-12-06 01:42:30ShuxinYangandBiaoLi
        Communications in Theoretical Physics 2023年11期

        Shuxin Yang and Biao Li

        1 School of Mathematics and Statistics,Ningbo University,Ningbo 315211,China

        2 School of Foundation Studies,Zhejiang Pharmaceutical University,Ningbo 315500,China

        Abstract The dressing method based on the 2 × 2 matrix -problem is generalized to study the complex modified KdV equation (cmKdV).Through two linear constraint equations,the spatial and time spectral problems related to the cmKdV equation are derived.The gauge equivalence between the cmKdV equation and the Heisenberg chain equation is obtained.Using a recursive operator,a hierarchy of cmKdV with source is proposed.On the basis of the -equation,the N-solition solutions of the cmKdV equation are obtained by selecting the appropriate spectral transformation matrix.Furthermore,we get explicit one-soliton and two-soliton solutions.

        Keywords: complex modified KdV equation,-dressing method,Lax pair,soliton solution

        1.Introduction

        In this paper,we consider the complex modified Korteweg–de Vries equation (cmKdV) [1] as follows:

        Here,u=u(x,t) is a complex function of x and t.The cmKdV equation is a typical integrable partial differential equation.It has been studied extensively over the last decades and its mathematical properties are well-documented in the literature.For example,the soliton solutions,the breathers and superregular breathers,the rogue wave solutions,the existence and stability of solitary wave solutions,periodic traveling waves,exact group invariant solutions and conservation laws have been discussed in [1–11].On the other hand,it is often suitable for many physical situations,such as the propagation of few-cycle optical pulses in cubic nonlinear media,electromagnetic wave propagation in liquid crystal waveguides and transverse wave propagation in molecular chain models [12–17].

        The layout of this paper is organized as follows.In section 2,we obtain the Lax pair with singular dispersion relation by using the-dressing method.In section 3,we introduce a cmKdV hierarchy with source,which contains a cmKdV hierarchy for a special case.In section 4,the Nsoliton solutions is constructed.As an application of the Nsoliton formula,we discuss one-soliton and two-soliton in section 5.Finally,the conclusions will be drawn based on the above sections.

        2.Spectral problem and Lax pair

        2.1.The spatial spectral problem

        We consider the 2 × 2 matrix-problem in the complex kplane

        with a boundary condition ψ(x,t,k) →I,k →∞,then a solution of the equation (2.1) can be written as

        where Ckdenotes the Cauchy–Green integral operator acting on the left.The formal solution of-problem (2.1) will be given from (2.2) as

        For convenience,we define a pairing [26]

        which can be shown to possess the following properties

        It is easy to prove that for some matrix functions f (k) and g(k),the operator Cksatisfies

        It is well known that the Lax pairs of nonlinear equations play an important role in the study of integrable systems.Such as the Darboux transformation,inverse scattering transformation,Riemann–Hilbert method,and the algebro-geometric method rely on their Lax pairs.Here we prove that if the transform matrix R(x,t,k) satisfies a simple linear equation,the spatial-time spectral problems of the cmkdv equation can be established from (2.1).In particular,we obtain the spatial-time spectral problems of the cmKdV equation.

        Proposition 1.Let the transform matrix R satisfies

        whereσ3=diag (1,-1),then the solution ψ of the-equation (2.1) satisfies the following spatial spectral problem

        Proof.Using (2.3) and (2.6),we get

        According to the definition of Ck,we can obtain

        SinceRCk=I-I· (I-RCk),then we fnid

        Substituting (2.10) and (2.11) into (2.9),we obtain

        From (2.10),we can get Substituting (2.13) into (2.12),we have equation (2.7).

        2.2.The time spectral problem

        Proposition 2.Suppose that R satisfies the linear equation

        which comprises both a polynomial part Ωp(k) and a singular part Ωs(k) andω(ξ) is a scalar function.Then the solution ψ of the-equation (2.1) leads to time spectral problem

        Remark 1.Asω(k)=0,the equation (2.16) reduces to

        which together with (2.7) gives the Lax pair of the cmKdV equation (1.1).

        Proof.We first use the polynomial dispersion relation only Ω=Ωp=4ik3σ3.From equations (2.2),(2.3) and (2.15),we arrive that

        Through the following direct computation

        then (2.18) is changed to

        By using (2.6),(2.7) and (2.8),we obtain

        Hence,equation (2.20) reduces to

        By means of (2.6),(2.7) and (2.8),we have

        Substituting (2.23) into (2.22) leads to the time evolution equation

        In the following,we consider the singular dispersion relation in (2.15).In the same way,we have

        And resorting (2.2) and (2.5),ψR(shí)ΩsCkin (2.25) satisfeis

        Hence,we have

        By using the relations

        we find that

        by which,then (2.27) gives a time-dependent linear equation with the singular dispersion relation

        which together with (2.24) gives (2.16).

        2.3.Gauge equivalence

        In this section,we prove that there is a gauge equivalence between the cmKdV equation and the Heisenberg chain equation.

        Proposition 3.The cmKdV equation (1.1) is gauge equivalent with the Heisenberg chain equation

        Proof.Making a reversible transformationg(x,t),

        Then we can calculate thatφ(x,t,k) satisfeis the following-problem,

        The equation (2.30) admits a solution

        From (2.31),we have

        Deriving equation (2.32) with respect to space variable x and using (2.7),we obtain

        Similar to the previous calculation,we can simplify the above formula to

        We can choose the function g to satisfy the following condition

        LetS=g-1σ3g,then the equation (2.33) gives the spatial–spectral problem

        The compatible condition leads to the Heisenberg chain equation

        3.Recursive operators and cmKdV hierarchy

        In this section,we derive the cmKdV equation with a source.First of all,we define the matrix M in the following form

        By using (2.8) and (3.1),we can obtain the following proposition.

        Proposition 4.Q defined by (2.8) satisfies a coupled hierarchy with a source M

        Remark 2.For the special case whenn=2,αn=2i,the hierarchy (3.2) give the cmKdV equation with source

        Proof.Differentiating the expression of Q with respect to t yields

        Because off(k)Ck=f(k),then we have

        By using (3.5),we can obtain

        which leads to

        Therefore,using (2.4) and (2.14),equation (3.6) can be simplified to

        Taking into the fact that Ωp=αn knσ3,αn=const and Ωs→ 0ask→∞,the above equation can be further reduced

        By virtue of the spectral problem (2.7),one can verify that

        From (3.9),they satisfy the following equations

        which lead to

        where

        The operator Λ usually is called as recursion operator.We expand (Λ -k)-1in the series

        Substituting it into (3.8) leads to the equation (3.2).

        4.N-soliton solutions of cmKdV equation

        In this section,we will derive the N-soliton solutions of the cmKdV equation (1.1).

        Proposition 5.Suppose thatkjandare 2N discrete spectrals in complex plane C.we choose a spectral transform matrix R as

        where cjis const andθ(k)=kx+4k3t,then the cmKdV equation (1.1) admits the N-soliton solutions

        whereMaugis(N+1) ×(N+1) matrices defnied by

        Proof.Substituting (4.1) into (2.8),yields

        Replacing k in (4.4) with kn,and k in (4.5) with,we get a system of linear equation

        Figure 1.One-soliton solution of equation (1.1) with ξ=0.2,η=0.3,ω1=-2,ω2=0.3,ξ0=φ0=0.

        The above equation can be rewritten as

        where M is N × N matrices defined by

        According to Cramer’s rule,the equation has the following solution

        Finally,substituting (4.8) into (4.3) further simplifies it to (4.2).

        5.Application of the N-soliton formula

        In the following,we will give the one-soliton and two-soliton solutions for the cmKdV equation (1.1).

        ?For N=1,taking k1=ξ+iη,the formula (4.2) gives the one-soliton solution of the cmKdV equation (1.1)

        where α=(-3ξη +(ξ3+ω1)/η)t+ξ0,φ=-2(3ξ2η -η3)t+2ω2t+φ0.The graphic of the one-soliton solution is shown in figure 1.

        Figure 2.Two-solion solution of equation (1.1) with k1=-0.1+0.2i,k2=-0.2+0.4i,c1=1+4i,c2=-1+i.

        ?For N=2,the formula (4.2) gives the two-soliton solution of the cmKdV equation (1.1) that is given by

        with viand vjbeing two arbitrary constants.The graphic of the two-soliton solution is shown in figure 2.

        6.Conclusion

        Acknowledgments

        This work is supported by the National Natural Science Foundation of China (Grant No.12175111,11975131),and the KC Wong Magna Fund in Ningbo University.

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