Ying-Yang Qiu(邱迎陽),Jing-Song He(賀勁松),and Mao-Hua Li(李茂華)

Department of Mathematics,Ningbo University,Ningbo 315211,China

Abstract The N-fold Darboux transformation(DT) of the nonlinear self-dual network equation is given in terms of the determinant representation.The elements in determinants are composed of the eigenvalues λj(j=1,2...,N)and the corresponding eigenfunctions of the associated Lax equation.Using this representation,the N-soliton solutions of the nonlinear self-dual network equation are given from the zero “seed” solution by the N-fold DT.A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λj → λ1and by using the higher-order Taylor expansion.For 2-degenerate and 3-degenerate solitons,approximate orbits are given analytically,which provide excellent fit of exact trajectories.These orbits have a time-dependent “phase shift”,namely ln(t2).

Key words:nonlinear self-dual network equation,Darboux transformation,soliton,degenerate solution

1 Introduction

The studies of the discrete integrable systems were initiated in the middle of 1970s.Hirota had discretized various integrable equations such as the nonlinear partial difference KdV equation,[1]the discrete-time Toda equation,[2]the discrete Sine-Gordon(SG)equation,[3]the Liouville’s equation,[4]and the Bcklund transformation of the discrete-time Toda equation,[5]based on the bilinear transformation methods.Following those works,the discrete nonlinear systems may be used to the diverse areas to describe such physical situations as the rogue waves in optical fibers and water tanks,[6]and the general rogue waves in the focusing and defocusing Ablowitz-Ladik equations.[7?8]In the ladder type electric circuit,connection type electric circuit,in connection with the propagation of electrical signals is the following nonlinear self-dual network equation[9?10]

where In=I(n,t)and Vn=V(n,t)are voltage and current in the n-th capacitance and inductance respectively,the functions of the discrete variable n and time variable t,In,t=dIn/dt,and Vn,t=dVn/dt.

During the recent decades,finding analytical and explicit solutions are one of the most aspect in the studies of discrete systems[11?14]There are quite a few methods of the nonlinear integrable systems,such as the inverse scattering transformation,the bilinear transformation methods of Hirota,the dressing method,the B¨acklund and the Darboux transformation(DT),the algebraic curve method.[15?17]It is known that the DT[17]is a powerful tool not only for the continuous integrable system,which generates new solutions of an integrable equation from a “seed” solutions,but also is useful for the discrete integrable system.Matveev gave the discrete DT of the differential-difference equation.[18]From then on the various DT of the discrete integrable system are becoming more and more important for the discrete problems.[19?20]In recent years,the determinant representation TNis widely used to get degenerate solitons and breathers[21?23]by a limit λj→ λ1,and even get rogue waves by a double degeneration with the help of limit λj→ λ1→ λ0.[24?26]Although there are some known solutions[27]of Eq.(1),the determinant representationthe N-fold DT has not been given in literature.This fact inspires us to constructof Eq.(1).

The study of the degenerate soliton can go back to Zakharov who has shown that the distance of two peaks in a 2-degenerate soliton increases with time like ln(4η2t)as|t|→ ∞.[28]The positons[29?30]of KdV was firstly introduced by Matveev,which is a special degenerate soliton involved a positive eigenvalue.The positon solutions have many interesting properties that different from the other of soliton solutions.Then the positon solutions have been constructed for many models,such as the defocusing mKdV equation,[31?32]the SG equation[33]and the Todalattice.[34]Moreover,positon can also be given by Hirota method and the limit of eigenvalues λj→ λ1.[35]It is easy to see from Refs.[28]–[36]that the essential feature of the above solutions is the degeneration of eigenvalues.This fact motivates us to construct the N-degenerate soliton of the nonlinear self-dual network equation by degenerate limit of eigenvalues λj→ λ1(j=2,3,...,N)according to theand further discuss its properties.

This paper is organized as follows.In Sec.2,a determinant representationof the N-fold DT for the nonlinear self-dual network equation is given.The new solutions ofandgenerated byare also provided explicitly by determinants.In Sec.3,the N-degenerate solitons In?dsand Vn?dsare obtained from N-solitons by a limit λj→ λ1,namely the degeneration of eigenvalues.Our conclusions and discussions are provided in the last section.

2 Determinant Representation of DT

First,the Lax pair of the nonlinear self-dual network equation is as follows[27]

where ψn=(ψ11,ψ12)Tis the vector eigenfunction,λ is the eigenvalue parameter independent of n and t,the shift operator E is defined by Ef(n,t)=f(n+1,t)=fn+1,n∈Z,t∈R.From the compatibility condition of the nonlinear self-dual network equation ψn,t= ψt,n,we can obtain the zero curvature equation

It is easy to find that Eq.(4)is compatible with the Eq.(1)by the calculations.Next,the discrete DT for Eq.(1)is given on the basis of Eqs.(2)and(3).So,we introduce an N-fold gauge transformation such that

Let N=1,set one-fold DT as

Here eigenfunction ψ1is in the form of

Solving Eq.(10)yields

where the matrix

where

In this case,we are going to make sure that each element of the matrix is a determinant consisting of the same submatrix W1.The representation given in Eq.(11)is more helpful to understand clearly and accurately the process for iteration ofin order to get the N-fold DT.It is clear that

Solving the algebraic equations given by the its kernel,i.e.,then

Here

which can be determined by solving the algebraic equationsThus we get following theorem.

Theorem 1The determinant representation of the N-fold DT is expressed by

Here

Here

Corollary 1The N-fold DTgenerates two new solutions from initial“seed” solutions Inand Vn,namely

Here

3 The N-Degenerate Solutions

In the last section,new solutionsandgenerated by theare given by the determinants involved with eigenvalue λj(j=1,2,...,N)and their eigenfunctions.Setting In=Vn=0 in Lax equations Eqs.(2)and(3),it is straightforward to get N-fold eigenfunctions

Taking In=Vn=0,and above eigenfunctions back into Corollary 1,it yields N-solitons

The aim of this section is to get N-degenerate solitons by taking a limit λj→ λ1in N-solitons.

Setting N=1,Eq.(17)produces two single solitons

Here α1=lnλ1. There are two kinds of soliton in Eqs.(18)and(19).

(i)When λ1>1,namely α1>0,theis a bright soliton(Fig.1(a))anda dark soliton(Fig.1(b)).

(ii)When 0< λ1<1,namely α1<0,theis a dark soliton andis a bright soliton.

The speed of two solitons is?α1/sinhα1.The trajec-tory ofis a line L01:tsinhα1+α1/2+nα1=0 in the(n,t)-plane,while L02:tsinhα1+ α1+nα1=0 for the.The height ofandare(e2α1? 1)/2eα1and?(e2α1? 1)/2eα1respectively,which can be confirmed by Fig.1 with λ1=3.

Fig.1 The amplitudes of single solitons and with λ1=3.(a)The amplitude is 4/3 for ;(b)The amplitude is?4/3 for

Setting N=2,Eq.(17)generates two 2-solitons:

Fig.2 (Color online)The discrete profiles of two-soliton solution in the(n,t)-plane with discrete variable n ∈ [?5,5].(a)Two bright soltions,α1= ?2,α2=1.5.(b)One bright and one dark solitons,α1=2,α2=3.(c)Two dark solitons,α1= ?1.5,α2=2.Here αj=lnλj,j=1,2.

Corollary 2The N-degengrate solitons of Eq.(1)are expressed by

Here

and ni=[(i+1)/2],[i]define the floor function of i.

Setting N=2,Corollary 2 yields a 2-degenerate solitonwhich is plotted in Fig.3.Note that Fig.3(b)is a density plot ofby using continuous variable n in order to get a good visibility,and all following density plots are generated by this way.To make a compact form of this paper,profiles ofare omitted.

Fig.3 (Color online)The discrete pro file of a two-degenerate solution in the(n,t)-plane(a)and its density plot(b)with α1=2.

Fig.4 (Color online)A sketchy demonstration of the limit λ2 → λ1in a two-soliton density plot)with a=0,c=1/2,λ1=2i/5,s1=0,and α1=2.From the left to the right,α2=3,2.5,2.2.Note that λj=eαj(j=1,2).(c)looks like Fig.3(b)very much.

We are now in a position to demonstrate intuitively the limit of degeneration by a graphical way based on analytical solutionsand,i.e.a 2-soliton approaches to a 2-degenerate soliton by λ2→ λ1.Note that we set αj=lnλj(j=1,2)and then use α2→ α1in order to show clearly this limit process.It is seen from Fig.4 that the trajectory(density plot)of a 2-soliton approaches to the trajectory(Fig.3(b))of a 2-degenerate soliton when α2goes to α1,which shows vividly the limit of a 2-soliton to a 2-degenerate soliton.Figure 4(c)is a good approximation of Fig.3(b)although?α= α2?α1=0.2 is not very small.Moreover,when|t|≥2,it is interesting to find in Fig.4 that the excellent agreement between the exact trajectories(density plots)and two approximate orbits,namely

which shows the validity of the approximate orbits(black lines).A remarkable feature of two approximate orbits is that there is a time-dependent “phase shift”,namely ln(t2). We are not able to decompose properly a 2-degenerate soliton into two single solitons,unlike we have obtained an excellent decomposition for the real mKdV and complex mKdV in Refs.[22–23],because the 2-degenerate solitonincludes a mixed combination of one bright soliton and one dark soliton,and there exists a transition from a bright soliton to a dark soliton(or a reverse process)along the time evolution.But we still retain this idea to find above orbits.

Fig.5 (Color online)The exact trajectories(density plots,red)and approximate orbits of a 2-degenerate soliton The approximate orbits L1(dashed line,black)and L2(solid line,black)in the(n,t)-plane with α1=2 when n ∈ [?10,10].

From Fig.6 the excellent agreement with exact trajectories(density plots)and three approximate orbits(black lines)for 3-degenerate soliton,namely

Note that there does not exist“phase shift”for the soliton propagating along

Fig.6 (Color online)The exact trajectories(density plots,red)and approximate orbits of a 3-degenerate soliton with α1=2 when n ∈ [?15,15].The approximate orbits (dashed line,black), (solid line,black)and (dot line,black)in the(n,t)-plane.

4 Summary and Discussion

In this article,a determinant representation of the N-fold Darboux transformationof the nonlinear selfdual network equation is given in Theorem 1.It is easy to verify(λj)ψj=0(j=1,2,...,N),which shows ψj(j=1,2,...,N)is the kernel of.Using this representation,the N-soliton solutions are obtained in Eq.(17).A general form of the N-degenerate solitons is given in Corollary 2,which is obtained by a limit λj→ λ1and by using higher-order Taylor expansion from the N-solitons.For 2-degenerate and 3-degenerate solitons,approximate orbits(Li,?Lj(i=1,2;j=1,2,3))are given analytically,which provide excellent fit of exact trajectories(density plots),see Figs.5 and 6.A remarkable feature of approximate orbits in the degenerate solitons is that there is a time-dependent “phase shift”,namely ln(t2).

If we compare our results with the work in Ref.[27]of the self-dual network equation,our results have the following advantages and innovation points.

(i)The determinant expressionis given for the first time.

(ii)The one-soliton solution is analyzed to classify the bright and dark soliton,and its height,speed and trajectory are presented analytically.

(iii)The N-degenerate solutions are given by determinants.The approximate orbits of 2-degenerate and 3-degenerate solitons and the time-dependent “phase shift”are presented in analytical way.

(i)The former has a different form as a polynomial of λ due to the new relations of matrix coefficients,see Eq.(12).

(ii)Here we just replace one column in WNto get new solutions by one same column vector in former,see Eqs.(14)and(15),while two column vectors are needed for a continuous system in Ref.[38].

Thus it is highly nontrivial to extend determinant representation of N-fold DT to discrete system.

Finally,the explicit form ofN)can be given analytically by using Theorem 1,and thus provides a convenient tool to study the discrete soliton surfaces and the dynamical evolution ofin the near future.

Acknowledgments

We thank Prof.D.J.Zhang and Dr.X.Y.Wen for many helpful suggestions on this paper.