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.