The Hamiltonian Structures and Algebro-geometric Solution of the Generalized Kaup-Newell Soliton Equations

WEI Han-yu,PI Guo-mei

(1.College of mathematics and statistics,Zhoukou Normal University,Zhoukou,466001,China;2.College of Economics and Management,Zhoukou Normal University,Zhoukou,466001,China)

Abstract:Staring from a new spectral problem,a hierarchy of the generalized Kaup-Newell soliton equations is derived.By employing the trace identity their Hamiltonian structures are also generated.Then,the generalized Kaup-Newell soliton equations are decomposed into two systems of ordinary differential equations.The Abel-Jacobi coordinates are introduced to straighten the fl ows,from which the algebro-geometric solutions of the generalized Kaup-Newell soliton equations are obtained in terms of the Riemann theta functions.

Key words:Soliton equations;Hamiltonian structures;Algebro-geometric solutions;Riemann theta functions

§1.Introduction

As is well-known,it is very important to search for Hamiltonian structures and algebrogeometric solutions of soliton equations.On the one hand,algebro-geometric solutions reveal inherent structure mechanism of solutions of soliton equations and describe the quasiperiodic behavior of nonlinear phenomenon or characteristic for the integrability of soliton equations[1,2].On other hand,they can be used to find multi-soliton solutions,elliptic function solutions,and others[1,2].In a series of literatures[3-10],several systematic methods were developed from which algebro-geometric solutions for a lot of soliton equations have been obtained,such as the KP,mKP,nonlinear Schr?dinger,Boussinesq-Burgers,etc.Especially in Ref.[11],a new integrable hierarchy of evolution equations was obtained,the reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions,the mKdV equation and a generalized linear KdV equation.The algebro-geometric solutions of the generalized nonlinear heat equation were obtained by the use of theory on algebraic curves.In Ref.[12],Zhang presented a(1+1)-dimensional discrete integrable hierarchy with a Hamiltonian structure and generated a(2+1)-dimensional discrete integrable hierarchy,respectively.Then a new differential-difference integrable system with three-potential functions and its algebraicgeometric solution were derived.

In this paper,we will consider the following generalized Kaup-Newell soliton equations

If we take β=0,then the Eq.(1.1)can be reduced the classical Kaup-Newell soliton equations

In fact,the system(1.1)is the coupled integrable equations from the generalized Kaup-Newell soliton hierarchy,which allows the zero-curvature representation in the sense of Lax compatibility,the Hamiltonian structure in view of the trace identity[13,14].The main aim of this paper is to get Hamiltonian structures of the generalized Kaup-Newell hierarchy,and to obtain the algebro-geometric solution of the generalized Kaup-Newell soliton equations.

The whole paper is organized as follows.In the next section,we shall derive the hierarchy of generalized Kaup-Newell soliton equations,based on the trace identity,we construct the Hamiltonian structures of the generalized Kaup-Newell soliton hierarchy.In section 3,based on the Lax pairs of the generalized Kaup-Newell soliton equations,variable separation technique is used to translate the solution of the generalized Kaup-Newell soliton to solve ordinary differential equations.Then,a hyperelliptic Riemann surface of genus N and Abel-Jacobi coordinates are defined to straighten the associated fl ows.The jacobi’s inverse problem is discussed,from which the algebro-geometric solutions of the generalized Kaup-Newell soliton(1.1)are constructed in terms of the Riemann theta functions.

§2.The generalized Kaup-Newell soliton hierarchy and its Hamiltonian structures

To derive the generalized Kaup-Newell soliton hierarchy and its stationary hierarchy,we introduce the Lenard gradient sequence{Sj,j=0,1,2...}by

where Sj=(cj,bj,aj)and operators(? = ?/?x)

A direct calculation gives from the recursion relation(2.1)that

Consider the generalized Kaup-Newell spectral problem

and the auxiliary problem

with

where q,r are two potentials,and λ is a spectral parameter.Then the compatibility condition between(2.5)and(2.6)yields the zero curvature equation Utm-V(m)x+[U,V(m)]=0,which is equivalent to the generalized Kaup-Newell hierarchy

The second and third nontrivial equations in above hierarchy(2.7)are

and

Eq.(2.8)is our generalized Kaup-Newell soliton equations(1.1).

Next,we will construct the Hamiltonian structures of the generalized Kaup-Newell soliton hierarchy(2.7),let

where

Directly calculation gets

According to the trace identity[13,14],we have

Comparing the coefficients of λ-m,we obtain

we set m=0,and then get γ=-1.Therefore we establish the following equation

with

In speciality,the Hamiltonian functional of soliton equation(1.1)is

It follows now that the generalized Kaup-Newell soliton hierarchy in(2.7)possess the following Hamiltonian structures

It is direct to see a recursion relation

with

§3.Algebro-geometric solutions of the generalized Kaup-Newell soliton equations

Let φ =(φ1,φ2)Tand ψ =(ψ1,ψ2)Tbe two basic solutions of spectral(2.5)and(2.6),we use them to defines a matrix

in which f,g,h are three functions.A direct calculation shows that

which implies that the function detW is a constant independent of x and tm.Eq.(3.2)can be written as

and

We suppose that the functions f,g,h are finite-order polynomials in λ

Substituting(3.5)into(3.3)yields

It is easy to see that(3.6)implies

and the equation JG0=0 has the general solution

where α0is constant of integration.Therefore,if we take(3.8)as a starting point,then Gjcan be determined recursively by the relation(3.6).In fact,noticing kerJ={cS0|?c ∈ C}and acting with the operator(J-1K)kupon(3.8),we obtain from(3.6)and(2.1)that

where α0,α1,...,αkare integral constants.Substituting(3.9)into(3.6)yields a certain stationary evolution equation

where

this means that expression(3.5)are existent.

In what follows,equation(1.1)will be decomposed into two systems of solvable ordinary differential equations.Without loss of generality,let α0=1,from(2.1),(2.3)and(3.9),we have

By using(3.5),we can write g and h as the following finite products

By comparing the coefficients of the same power for λN-1that

Thus from(3.11)and(3.13),after a simple calculation,we get

Let’s consider the function detW,which is a(2N+2)th-order polynomial in λ with constant coefficients of the x- fl ow and tm- fl ow

Substituting(3.5)into(3.15)and comparing the coefficients of λ2N+1yields

which together with(3.11)gives

From(3.15)we see that

Again by using(3.11)and(3.12),we obtain

which together with(3.18)gives

In a way similar to the above expression,by using(2.6)(m=1,t1=t),(3.4),and(3.18),we arrive at the evolution of{uk}and{vk}along the tm- fl ow

Therefore,if the(2N+2)distinct parameters λ1,λ2,...,λ2N+2are given,and let uk(x,t)and vk(x,t)be distinct solutions of ordinary differential equations(3.20),and(3.21),then(q,r)determined by(3.14)is a solution of the generalized Kaup-Newell soliton equations(1.1).

In the following,we shall give the algebro-geometric solutions of the generalized Kaup-Newell soliton equations(1.1).We first introduce the Riemann surface Γ of the hyperelliptic curve

with genus N on Γ.On Γ there are two infinite points ∞1and ∞2,which are not branch points of Γ.We equip Γ with a canonical basis of cycles:a1,a2,...,aN;b1,b2,...,bN,which are independent and have intersection numbers as follows

We will choose the following set as our basis

which are linearly independent from each other on Γ,and let

It is possible to show that the matrices A=(Aij)and B=(Bij)are N×N invertible matrices[14,15].Now we defines the matrices C and τ by C=(Cij)=A-1, τ=(τij)=A-1B.Then the matrix τ can be shown to symmetric(τij= τji)and it has a positive-definite imaginary part(Im τ＞0).If we normalizeinto the new basis ωj

Then we have

Now we introduce the Abel-Jacobi coordinates as follows

which implies

with the help of the following equality

In a similar way,we obtain from(3.20)and(3.21)

On the basis of these results,we obtain the following

where γ(i)j(i=1,2)are constants,and

Now we introduce the Abel map A(p)

and Abel-Jacobi coordinates

According to the Riemann theorem[15,16],there exists a Riemann constant vector M∈CNsuch that the function

has exactly N zeros at u1,u2,...,uNfor m=1 or v1,v2,...,vNfor m=2.To make the function single valued,the surface Γ is cut along all ak,bkto form a simple connected region,whose boundary is denoted by γ.By Res.[15,16],the integrals

are constants independent of ρ(1)and ρ(2)with

By the residue theorem,we have

Here we need only compute the residues in(3.40).In a way similar to calculations in[17,18],we arrive at

where

with

Thus from(3.40)and(3.41),we arrive at

Substituting(3.42)into(3.14),then we get an algebro-geometric solution for the generalized Kaup-Newell soliton equations(1.1)

where A(t)and B(t)are arbitary complex functions about variable t.

§4.Conclusions

In conclusion,we propose a new generalized Kaup-Newell soliton hierarchy and its Hamiltonian structures.Based on the stationary generalized Kaup-Newell hierarchy,solutions of the generalized Kaup-Newell soliton equations are reduced to solving integrable ordinary differential equations.By introducing the Abel-Jacobi coordinates,we further obtain the algebro-geometric solutions of the generalized Kaup-Newell soliton equation.It should also be pointed out that the method used here is suitable for other soliton hierarchies,we will try to solve some other new integrable equations in the near future.

AcknowledgementsWe thank ZHANG Yan for helpful advices during the writing of this work.