Integrability,multi-soliton and rational solutions,and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter

Meng-Li Qin,Xiao-Yong Wenand Cui-Lian Yuan

School of Applied Science,Beijing Information Science and Technology University,Beijing 100192,China

Abstract Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter α abbreviated as RTL_(α)system by Suris,which may describe the motions of particles in lattices interacting through an exponential interaction force.First of all,an integrable lattice hierarchy associated with an RTL_(α)system is constructed,from which some relevant integrable properties such as Hamiltonian structures,Liouville integrability and conservation laws are investigated.Secondly,the discrete generalized(m,2N−m)-fold Darboux transformation is constructed to derive multi-soliton solutions,higher-order rational and semirational solutions,and their mixed solutions of an RTL_(α)system.The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis.Finally,soliton dynamical evolutions are investigated via numerical simulations,showing that a small noise has very little effect on the soliton propagation.These results may provide new insight into nonlinear lattice dynamics described by RTL_(α)system.

Keywords:RTL_(α)system,Hamiltonian structures,discrete generalized(m,2N-m)-fold Darboux transformation,soliton and rational solutions,asymptotic analysis

1.Introduction

Solitons are a class of nonlinear localized waves with particle-like properties[1].In the field of mathematical physics,the studies of soliton are mainly divided into continuous cases and discrete cases.Nonlinear partial differential equations(NPDEs)and nonlinear differential-difference equations(NDDEs)are usually used to describe the dynamical behaviors of solitons in continuous and discrete cases,respectively[1].A soliton in NDDEs is sometimes also referred to as a lattice-soliton[1].There have been many studies on continuous solitons in NPDEs which are used to describe some physical phenomena,such as nonlinear surface gravity waves propagating over a horizontal sea bed,propagation of a short wave in ferromagnets,optical logic switches and ultrashort pulse lasers,the nonlinear dynamics of optical solitons in nonlinear optics,shallow water wave,etc[2–7].Nevertheless,the discrete soliton in discrete NDDEs is still inadequate compared to their continuous part.In recent years,NDDEs,which are thought as spatially discrete counterparts of NPDEs,have received widespread attention because they may be used to describe various physical phenomena in different fields such as nonlinear lattice dynamics,pulses in biological chains,ladder type electric circuits,population dynamics,nonlinear optics,plasma physics and so on[1,8–14].Seeking discrete exact solutions,in particular discrete soliton solutions,plays a crucial role in understanding the physical phenomena described by NDDEs[8–14].Many effective methods for solving NDDEs have been proposed in the literature like the discrete inverse scattering transformation[12],discrete Hirota transformation[13,14],and discrete Darboux transformation(DT)[15–19]and discrete N-fold DT[20–26],etc.Among them,the discrete N-fold DT based upon Lax pair of integrable NDDEs is a powerful technique to get multi-soliton solutions without complicated iterative procedure[20–26].And recently,a generalized technique on the basis of the usual discrete N-fold DT has also been proposed[27–29].In comparison with the usual N-fold DT only providing the soliton solutions,the discrete generalized DT technique not only expresses the ordinary multisoliton solutions but also produces new higher-order rational solutions,semi-rational solutions and their mixed solutions.

One of the most famous and deeply researched integrable NDDEs is undoubtedly the Toda lattice which was originally discovered by Toda[1,8,9].Toda lattice may describe a lattice of particles interacting with nearest neighbors via forces exponentially depending on distances(see figure 1),and it is the first example of integrable NDDEs which also has important guiding significance in the study of nonlinear waves and ergode theory[1].Regarding Toda lattice and its relativistic version,some integrable properties have been studied such as Lax pairs[30–33],Hamiltonian structures[33–36],conservation laws[37],the exp-function method[38],the rotational expansion method[39],the tanh-method[40],Bäcklund transformation[41],DT[16,19,24,42]and so on.Toda lattice has an integrable generalization of relativistic version nowadays called the relativistic Toda lattice(RTL)with one perturbation parameter α proposed by Ruijsenaars[8,30,31]in the form of the following Newtonian equations of motion:

which may be viewed as a one-parameter perturbation of Toda lattice,where xn=x(n,t)is the function of the discrete spatial variable n and time variable t,in which n∈Z,t∈R,and α is a small parameter whose physical meaning is the inverse speed of light.When α→0 or α=0,equation(1)is just the usual Toda lattice.Introducing the following transformations

the time evolutions of anand bnare governed by the following equations of motion[8]:

where an=a(n,t)and bn=b(n,t)are the functions of discrete and time variables n and t respectively,andEquation(3)turns out to have much richer structure than the original Newtonian motion equation(1).Equation(3)is the‘minus first’flow of the RTL(α)hierarchy which is assigned an abbreviation as RTL_(α)system by Suris[8].Hereafter,we will use this abbreviated name for equation(1)or(3)which will be mainly discussed in this paper.Here we need to point out that RTL_(α)system(3)with the denominator studied in this paper is different from RTL systems without the denominator in[16,19,30–32,34–36,38–42].The 2×2 Lax representations for equation(3)and its discretization form have been given in[8].For the convenience of later discussions,we here list the 2×2 Lax pair for equation(3)as follows:

which is somewhat different from ones in[8],where λ is the spectral parameter independent of time t,E is the shift operator defined by Ef(n,t)=f(n+1,t),E-1f(n,t)=f(n-1,t),u=(an,bn+1)Tare the potential functions of variables n,t,andφn=(φn,ψn)T(T means transpose)is an eigenfunction vector.The compatibility condition Un,t=(EVn)Un-UnVnbetween the spatial part(4)and time evolution part(5)of Lax pair yields equation(3).To the best of the authors’knowledge,some integrable properties via the Tu scheme[33],various exact solutions via the discrete generalized(m,2N-m)-fold DT,and soliton dynamical behaviors via numerical simulations of equation(3)have not been reported in literature before.Different from the discrete generalized(m,N-m)-fold DT choosing the number m of different spectral parameter from 1 to N in[27–29],we will extend the discrete generalized(m,N-m)-fold DT to the discrete generalized(m,2N-m)-fold DT by selecting the spectral parameter number m between 1 and 2N so that we may give some new exact solutions.

Therefore,in this paper,we will extend the Tu scheme[33]and discrete generalized(m,2N-m)-DT to further study the integrability and exact solutions of equation(3).The rest of this article is organized as follows.In section 2,a discrete integrable lattice hierarchy associated with equation(3)is presented and some integrable properties including Hamiltonian structures and Liouville integrability will be discussed by using trace identity on the basis of the Tu scheme[33].Section 3 deals with infinitely many conservation laws of equation(3).Section 4 is devoted to the discrete generalized(m,2N-m)-fold DT of equation(3)based on its known Lax pair.In section 5,the exact solutions such as multi-soliton solutions,rational and semi-rational solutions and their mixed solutions for equation(3)are obtained by applying the resulting generalized DT,and their asymptotic states are analyzed.In section 6,the dynamical behaviors of multi-soliton solutions will be discussed via numerical simulations.Some conclusions and discussions are given in section 7.

2.A RTL_(α)hierarchy and its Hamiltonian structures

For the discrete 2×2 matrix spectral problem(4),we will use the Tu scheme to construct the lattice hierarchy of(3).Solving the following stationary discrete zero-curvature equation

in which

results in the following recursion relations

Ifis defined by

from equation(6)together with(7),we arrive at

Since

we can see that(9)and(10)have different forms,in order to derive the desired lattice hierarchy,we need to seek an appropriate modification matrix δnto modify.It is easy to verify if we take

then

Assuming that the time evolution of φnmeets the equationsthen the compatibility conditionimplies

which yields the following integrable lattice hierarchy:

(1)When m=0,the lattice hierarchy in equation(15)reduces to RTL_(α)system(3)with t0=t as

whose time part of Lax pair is

which is a little different from equation(5).In fact,they are equivalent under gauge transformation.

(2)When m=1,from the lattice hierarchy in equation(15),we arrive at

whose time part of Lax pair is

in which

Here equation(18)is called the second-order RTL_(α)system which is not discussed in this paper.By continuing this process,we can get a series of discrete integrable NDDEs.

Next we will construct the Hamiltonian structures of equation(15).Before that,we first look back to some symbols used in this paper[33].The variational derivative of the scalar function fnwith regard to uiis defined asThe formuladenotes the inner product between vector functionsandThe Poisson bracket[33]for the Hamiltonian operator J between functions fnand gnis defined byThe operator J*defined by(fn,J*gn)=(Jfn,gn)is called the adjoint operator of J with respective to the inner product,in which J is described as the skew-symmetric operator if J=-J*.

Next,we define＜U,V>=tr(UV),where U and V are arbitrary square matrices.If we set

then we have

By using the trace identity[33]

we arrive at

Comparing the coefficients of λ-2m-3on both sides of equation(21),we have

Setting m=0,from equation(24)we have ε=0.Letthen

in which

Hence,equation(24)can be rewritten as

Hence we have successfully written the lattice hierarchy(18)in the above Hamilionian form(28).It can be verified that J and Jη are skew-symmetric operators.The Hamiltonian functions(m≥0)described by equation(25)are pairwise involutory concerning Poisson bracket.So we have the following theorem:

Theorem 1.The hierarchy(15)or the Hamiltonian form(28)possesses Liouville integrability.

3.Conservation laws of equation(3)

In the previous section,we have investigated Liouville integrability of equation(3).In this section,we study its infinitely many conservation laws which will further verify the integrability of equation(3)[1,37,43].

and

From the time part(17)of Lax pair in the previous section,we have

From(29)and(31),we can derive the following conservation laws for equation(3)as follows:

Equating the same powers of λ on both sides of equation(32),we obtain infinitely many conservation laws for equation(3).The first three conservation laws are listed as follows

with

where Tkand Xkdenote the conserved densities and associated fluxes respectively.are motion constants,andandin physical meanings represent the total momentum and total energy of the lattice respectively.The existence of infinitely many conservation laws means that equation(3)is a discrete integrable system.

4.Discrete generalized(m,2N−m)-fold DT of equation(3)

In this section,we will construct the discrete 2N-fold DT of equation(3),and then extend it to the discrete generalized(m,2N-m)-fold DT.We first introduce the following gauge transformation:

wheresatisfies the Lax pair(4)and(17),i.e.

in which the number N is a positive integer,andand(j=1,2,…N)are functions of the variables n and t which are determined by the linear algebraic system T(λi)φi,n(λi)=0(i=1,2,…,2N),whereare 2N solutions of Lax pair(4)and(17)for 2N spectral parameters λi.When the 2N distinct parameters λi(λi≠λj,i≠j)are suitably chosen so that the determinant of the coefficients of 4N functions,and(j=1,2,…N)is nonzero.Hence the Darboux matrix Tnin(36)can be uniquely determined.

According to the previous analysis,we come to the following 2N-fold DT theorem of equation(3):

Theorem 2.Letφi,n(λi)=(φi,n,ψi,n)Tbe 2N column vector solutions of Lax pair(4)and(17)for the spectral parametersλi(i=1,2,…,2N)with the initial solutionsan,bnof equation(3),then the 2N-fold DT of equation(3)between the old solutionsan,bnand the new solutions˜nis given by

where

in which

Proof..Letand

By direct calculation we know thatf11(λ,n)is the(4N+2)th order polynomial in λ,f12(λ,n),f21(λ,n)are the(4N+1)th order polynomials in λ,andf22(λ,n)is the(4N)th order polynomial in λ.

Through direct calculations,we can verify that λi(i=1,2,…,2N)are the roots of fj,k(λ,n)(j,k=1,2).Therefore,we can rewrite(39)as

with

Hence we have

Compare the coefficient of λ on both sides of equation(43)together with(36),we can get the following results as

Thus,we havePn=.In other words,the matrices Unandhave the same forms under the transformations(34)and(37).

Next,we try to prove that the matrixhas the same form asunder the transformations(34)and(37).Letand

Through a straightforward calculation,we know that g11(λ,n),g22(λ,n)are(4N+2)th order polynomials in λ,g12(λ,n),g21(λ,n)are(4N+1)th order polynomials in λ.

from which we can verify that g11(λi,n),g12(λi,n),g21(λi,n)and g22(λi,n)are all zeroes,so we have

with

Therefore we obtain

Expanding and comparing the coefficient of λ on both sides of equation(48)together with(36),we can get the following results as

The transformations(34)and(37)using 2N spectral parameters λi(i=1,2,…,2N)are usually called the 2N-fold DT of Lax pair(4)and(17)of equation(3).Here the number 2N represents the order of DT.For the 2N-fold DT,we need 2N spectral parameters to construct multi-soliton solutions.In[27–29],one of the authors in this paper has presented a discrete generalized(m,N-m)-fold DT with no more than N spectral parameters(i.e.1≤m＜N)to give the rogue wave and rational soltion solutions of several integrable NDDEs.Next,we shall extend this idea to equation(3)for seeking some new rational,semirational and mixed solutions by using no more than 2N spectral parameters.To this aim,we must reduce the number of spectral parameter λ,in other words,we use the less m(1≤m＜2N)spectral parameters.We know that the condition Tn(λi)φn(λi)=0(i=1,2,…,m)yields the linear algebraic system with only 2m algebraic equations so that we can not determine 4N unknown variablesTo obtain the linear algebraic system with 4N algebraic equations of 4N unknown variablesfor every λi,we expand

from which the determinant of the coefficients for system(4)is nonzero when the m spectral parameters λiare suitably chosen so thatin the Darboux matrix Tnare uniquely determined by(4).Moreover,theorem 2 still holds for the Darboux matrix Tnwith newgiven by the new system(4).Due to new distinct functionsobtained in the Darboux matrix Tnin theorem 2,we can derive the new DT with m spectral parameters λi.Here the transformations(34)and(37)with m spectral parameters related to new functionsgiven by(4)are called the discrete generalized(m,2N-m)-fold DT of equation(3)which is summarized as the following theorem:

Theorem 3.Letφi,n(λi)=(φi,n,ψi,n)Tbe m column vector solutions of Lax pair(4)and(17)for the spectral parametersλi(i=1,2,…,m)with the initial solutionsan,bnof equation(3),then the generalized(m,2N-m)-fold DT of equation(3)from the old solutionsan,bnto the new solutions˜nis given by

where

Remark 1.Here the transformations(34)and(50)are called the discrete generalized(m,2N-m)-fold DT of equation(3).Note that m in the notation‘(m,2N-m)’denotes the number of the distinct spectral parameter we use,2N denotes the order number of DT.In theorem 3,vimeans the order number of the highest derivative in Taylor series expansion for everyand2N-m=is the order number sum of the highest derivative of Darboux matrix Tnor the vector eigenfunctionφi,n(λi).Notice that ifm=2Nandmi=0,theorem 3 can reduce to the discrete generalized(2N,0)-fold DT including the discrete 2N-fold DT when we do not make Taylor series expansion for everyφi,n(λi).If m=1 andmi=2N-1,theorem 3 can reduce to the discrete generalized(1,2N-1)-fold DT which is used to derive higher-order rational and semi-rational solutions of equation(3).If m=2 andmi=2N-2,theorem 3 can reduce to the discrete generalized(2,2N-2)-fold DT which is used to obtain mixed solutions of usual soliton solutions and rational or semi-rational solutions.If2＜m＜2N,theorem 3 can reduce to the other discrete generalized DTs which can give the new discrete mixed solutions.In the next section,the discrete generalized(m,2N-m)-fold DT withm=2N,1,2 will be used to derive usual soliton solutions,rational or semi-rational solutions and their mixed solutions for equation(3).

5.Explicit exact solutions and asymptotic state analysis of equation(3)

In this section,we will use the discrete generalized(m,2N-m)-fold DT with three cases m=2N,1,2 to discrete soliton solutions,discrete rational and semi-rational solutions,and their mixed solutions.

5.1.Multi-soliton solutions via the discrete generalized(2N,0)-fold DT

When m=2N,the discrete generalized(m,2N-m)-fold DT reduces to the discrete generalized(2N,0)-fold DT which includes the usual 2N-fold DT.If we do not make Taylor series expansion for every eigenfunction φi,n(λi)(i=1,2,…,2N),the discrete generalized(2N,0)-fold DT is just the usual 2N-fold DT,while if we do Taylor series expansion for one of eigenfunctions φi,n(λi)(i=1,2,…,2N),in fact,we can give some mixed solutions of usual soliton solutions and rational or semi-rational solutions.In this subsection,we shall use the usual 2N-fold DT to give multi-soliton solutions of equation(3).Substituting the initial seed solutions an=1,bn=0 into(4)and(17)gives one basic solution with λ=λi(i=1,2,…,2N)as follows:

with

According to(37),we can obtain exact 2N-soliton solutions of equation(3).To understand them,we plot their structures with N=1,2 as shown in figures 2–6.

(I)When N=1,λ=λi(i=1,2),based on the 2-fold DT in theorem 2,the transformation(37)produces the two-fold exact solutions as

where

with

The solutions(52)may be one-soliton or two-soliton solutions when the parameters are suitably chosen.

in which

with

The wave structures of one-soliton solutions(52)are shown in figure 2.Figure 2(a1)(b1)present the bell-shaped soliton structures of the componenton nonzero seed background.Figure 2(a2)(b2)display the anti-bell-shaped soliton structures of the componenton zero seed background.From figure 2,both one-soliton solutionsandpropagate stably with the same amplitude and remain their shapes and velocities unchanged during the propagation.

Figure 1.A one-dimensional lattice with fixed ends(see the first figure in[1]).

Case(b)When both of the two parameters λ1,λ2are not equal tothe solutions(52)are two-soliton solutions expressed as

where

with

To find whether the interaction between two solitons is elastic,we carry out the asymptotic analysis for solutions(54),which yields the following eight asymptotic expressions of.

Before the interaction t→-∞:

After the interaction t→+∞:

From the above analysis,we can see that the interactions between two solitons for the solutions(54)are elastic.When the parameters are properly chosen,the elastic interaction structures of two-soliton solutions(54)are shown in figure 3.Figure 3(a1)(b1)present the headon elastic interaction between two bell-shaped solitons structures of the componenton nonzero seed background.Figure 3(a2)(b2)display the head-on elastic interaction between one bell-shaped bright soliton and one anti-bell-shaped soliton of the componenton zero seed background.From figure 3,we can clearly see that the shapes of two solitons remain the same before and after the interactions such that their interactions are elastic,which are consistent with our asymptotic analysis above.

(II)When N=2,λ=λi(i=1,2,3,4),based on the 4-fold DT in theorem 2,the transformation(37)produces the fourfold exact solutions as follows:

where

with

When the parameters are suitably chosen,the solutions(55)may be three-soliton or four-soliton solutions whose corresponding evolution plots are shown in figures 4 and 5.Similar to two-soliton solutions’asymptotic analysis,we can also analyze the threesoliton and four-soliton solutions which are cumbersome and not presented here.Figure 4(a1)(b1)present the headon elastic interactions among two unidirectional overtaking bell-shaped solitons and one opposite bell-shaped soliton of the componenton nonzero seed background.Figure 4(a2)(b2)display the headon elastic interactions among two unidirectional overtaking antibell-shaped solitons and one opposite bell-shaped bright soliton of the componenton zero seed background.From figure 4,we can clearly see that three solitons preserve their shapes and amplitudes before and after the interactions.Figure 5(a1)(b1)present the headon elastic interactions among three unidirectional overtaking bell-shaped solitons and one opposite bell-shaped soliton of the componenton nonzero seed background.Figure 5(a2)(b2)display the headon elastic interactions among three unidirectional overtaking anti-bell-shaped solitons and one opposite bell-shaped bright soliton of the component bnon zero seed background.Note that we plot the absolute value of solutionfor the sake of a better view here.From figure 5,we can clearly see that four solitons keep their shapes and amplitudes before and after the interactions.In order to better understand the properties of equation(3)more comprehensively,we plot the structures for the combined potential termas illustrated in fgiure 6.From figure 6,we can distinctly see that the combined potential termpresents the stable one-,two-,three-and four-soliton structures which also shows that the total energy of the system is conserved from another aspect.

Remark 2.The point here is that the solutions(37)are the even(2N)-soliton solutions if allλj(j=1,2,…,2N)are not equalhowever,if only one ofλj(j=1,2,…,2N)is equivalent tothe solutions(37)are the odd(2N-1)-order soliton solutions.The multi-soliton solutions of equation(3)are composed of the even(2N)-soliton and odd(2N-1)-soliton solutions which is a very interesting feature worthy of further study.

5.2.Rational and semi-rational solutions via the discrete(1,2N−1)-fold DT

In this subsection,we will use the discrete generalized(1,2N-1)-fold DT with singl e eigenvalue to investigate some rational solutions and semi-rational solutions of equation(3)when m=1.First of all,we fix the spectral parameter λ=λ1+ε.Then expanding the vector function φ1,nin(51)as two Taylor series around ε=0 and choosingwe arrive at

where

with

(I)When N=1,based on the discrete generalized(1,1)-fold DT in theorem 3,we can obtain the first-order rational solutions of equation(3)as

in which

with

Through a direct calculation,the simplification analytical expressions of solutions(57)are listed as follows:

from which we can see thatpossesses singularity at two paralleled straight lines 10n+5t-8=0 and 10n+5t-2=0,whilehas singularity at two paralleled straight lines 10n+5t-8=0 and 10n+5t-12=0.Moreover,we can conclude thatas n→±∞or t→±∞.

It is important to point out that we have derived the previous rational solutions composed of the polynomials of variables n,t if we expand φ1,nin(51)aroundWhenthrough the similar process like above,we can obtain the semirational solutions which are made up of polynomial functions and exponential functions.For instance,we fix the spectral parameter λ=λ1+ε with λ1=3 in(51),based on the generalized(1,1)-fold DT,from(57)we can derive the first-order semirational solutions whose simplification forms are listed as follows:

where

From the above expressions,we can clearly see that the solutions(59)consist of polynomial functions and exponential functions.Here we call this kind of solutions the semi-rational solutions relative to the rational solutions.With the help of symbolic computation,we can easily verify that the solutions(59)are correct by inserting them into equation(3).

(II)When N=2,based on the generalized(1,3)-fold DT in theorem 3,we can get the second-order rational solutions of equation(3)as

where

with

The simplification forms of solutions(60)are precisely expressed as

where

Remark 3.It can be seen thatandpossess singularity at four curves5ξ1-2 =0,5ξ1-8 =0,5ξ2-2=0,5ξ2-8 =0,which also are the four center trajectories of solution,whileandpossess singularity at four curves 5ξ1-12 =0,5ξ1-8 =0,5ξ2-12 =0,5ξ2-8 =0,which are also the four center trajectories of solutionFrom the asymptotic expressions(62)and(63)we can clearly that the asymptotic expressions of second-order rational solutions are formally consistent with the first-order rational solutions,but the main difference is that the trajectories of the first-order rational solutions are straight lines,whereas the trajectories of the higher-order rational solutions are curves.

If we expand φ1,nin(51)around λ1=3,based on the generalized(1,3)-fold DT,from(60)we can derive the second-order semi-rational solutions whose simplification analytical forms are very cumbersome,and so not presented here.

(III)When N=3,based on the discrete generalized(1,5)-fold DT in theorem 3,we can get the third-order rational solutions of equation(3)as

where

with

in which

It should be noted that the analytical expressions of(64)are also very complicated and not listed here.If we expand φ1,nin(51)around λ1=3,based on the generalized(1,5)-fold DT,from(64)we can also derive the third-order semi-rational solutions whose simplification analytical forms are very cumbersome,and so not presented here.

Next,we summarize some mathematical features of the previous discrete rational solutions for equation(3)listed in tables 1 and 2.In two tables,the first column shows the order number of the rational solutions,the second and third columnsshow the highest power in the numerator and denominator polynomials involved in the solution anrespectively,the fourth and fifth columns show the highest power in the numerator and denominator polynomials involved in the solution bnrespectively,the sixth column means the background level of the solution an,while the last column provides the background level of the solution bn.From table 1,we can easily that for the rational solution anof order j,the highest powers in the numerator and denominator polynomials are both 2j(2j-1),while for the rational solution bnof order j,the highest powers in the numerator and denominator polynomials are 2j(2j-1)-2 and 2j(2j-1)respectively.

Table 1.Main mathematical features of rational solutions an and bn of order j.

Table 2.Main mathematical features of rational solutions an and bn of order j.

It is particularly worth pointing out that we can derive some new rational solutions from(57),(60)and(64)if we fix the spectral parameter λ=λ1+ε withand choosein(51)and(56).Moreover,we here omit their analytical expressions and only sum up their mathematical properties listed in table 2.

5.3.Mixed solutions via the discrete generalized(2,2N−2)-fold DT

In the previous two subsections,we have used the discrete 2N-fold DT with 2N spectral parameters to derive multi-solutions of equation(3),and also used the discrete generalized(1,2N-1)-fold DT with only one spectral parameter to derive the rational and semi-rational solutions of equation(3).In this subsection we will employ the discrete generalized(2,2N-2)-fold DT with two spectral parameters to give some mixed solutions of usual soliton solutions and rational or semi-rational solutions of equation(3).To save space,we only discuss the discrete generalized(2,2)-fold DT(i.e.the discrete generalized(2,2N-2)-fold DT with N=2).In what follows,we will only list a type of mixed solution of usual one-soliton solution and rational solution via the discrete generalized(2,2)-fold DT.

First of all,when N=2,we need to use two spectral parameters,here we set the parametersand(e.g.λ2=3),then we let the spectral parameter λ in equation(51)as λ=λ1+ε,and expand the vector function φi,nin(51)as Taylor series around ε=0 by choosing C11=C12=C21=C22=1,based on the discrete generalized(2,2)-fold DT,we can get the mixed solutions of usual one-soliton solution and rational solution as

with

Remark 4.The mixed solutions(65)are also composed of polynomial functions and exponential functions which completely differs from the semi-rational solutions(59).The mixed solutions(65)need to use two spectral parameters,while the semirational solutions(59)only need one spectral parameter.With the help of symbolic computation,one can verify the mixed solutions(65)by substituting them into equation(3),and we will not draw their evolution structures due to their singularity here.Similarly,some more complicated mixed solutions can be given by means of the discrete generalized(2,2N-2)-fold DT whenN>2,whose simplified expressions are too complicated and not presented here.

6.Dynamical behaviors of soliton solutions

In this section,we use numerical simulations to illustrate the dynamical behaviors of the previous soliton solutions of equation(3)by using finite difference method[44].Figure 7–9 exhibit dynamical behaviors of the exact one-,two-,and threesoliton solutions respectively.In figures 7–9,it is notable that the first columns show the exact soliton solutions corresponding to figures 2–4 respectively,the second columns present the numerical solutions with no noise by means of exact soliton solutions as initial conditions of the difference scheme algorithm,while the last two columns present the perturbed numerical solutions through adding 2% and 8% small noises to the exact solutions as initial conditions respectively.

Figure 7.(Color online)One-soliton solutions(52)with the same parameters as figure 2.(a1)(a2)Exact solutions.(b1)(b2)Numerical solutions with no noise.(c1)(c2)Numerical solutions with a 2% noise.(d1)(d2)Numerical solutions with a 8% noise.

Figure 8.(Color online)Two-soliton solution(54)with the same parameters as figure 3.(a1)(a2)Exact solutions.(b1)(b2)Numerical solutions with no noise.(c1)(c2)Numerical solutions with a 2% noise.(d1)(d2)Numerical solutions with a 8% noise.

Figure 9.(Color online)Three-soliton solutions(55)with the same parameters as figure 4.(a1)(a2)Exact solutions.(b1)(b2)Numerical solutions with no noise.(c1)(c2)Numerical solutions with a 2% noise.(d1)(d2)Numerical solutions with a 8% noise.

From figures 7–9(a1)(b1)–(a2)(b2),we can clearly see that the wave propagations of soliton solutions with on noise are almost identical to the corresponding exact soliton solutions in each case which also show the accuracy of our numerical scheme.When a 2% small noise is added to both the initial exact solutions,the time evolution profiles are also almost their same as the corresponding exact soliton solutions in a short time(see figures 7–9(c1)–(c2)).However,if a 8%noise is added to the initial exact solutions,the wave propagations only have a weak oscillation at a later time(see figures 7–9(d1)–(d2)).Numerical results in figures 7–9 show that these soliton solutions have stable evolutions against a small noise.

7.Conclusions and discussions

In this paper,we have studied equation(3)which may describe particle vibrations in lattices with an exponential interaction force.First of all we have constructed a discrete integrable hierarchy(15)related to equation(3),from which some related properties such as Hamiltonian structures,Liouville integrability and infinitely many conservation laws have been discussed.Secondly,the discrete generalized(m,2N-m)-fold DT(34)and(50)of equation(3)has been constructed in detail to derive its multi-soliton solutions with m=2N case,rational and semi-rational solutions with m=1 case and their mixed solutions with m=2 case.The asymptotic state analysis for multi-soliton solutions and rational solutions are investigated in detail,from which some elastic interaction phenomena also have been discussed(see figures 2–6).Moreover,we find that the even soliton solutions may reduce to odd soliton solutions by choosing special spectral parameters,and together they make up the multisoliton solution of equation(3).We also have summarized a few mathematical features of different-order rational solutions of equation(3)(see tables 1 and 2).At Last,numerical simulations are utilized to illustrate the dynamical behaviors of one-,twoand three-soliton solutions,showing that the evolutions of such soliton solutions are robust against a small noise(see figures 7–9).The results given in this paper might provide a few new views about understanding the particle vibrations in lattices described by equation(3).

In theory,we can use the discrete generalized(m,2N-m)-fold DT to give more new exact solutions of equation(3).But in fact,the calculations are quite complicated,and how to discuss these novel results is worthy of further investigation in this research area.We believe that the key technique used in this paper can also extended to solve other Lax integrable NDDEs.

Acknowledgments

This work is supported by National Natural Science Foundation of China(Grant No.12 071 042)and Beijing Natural Science Foundation(Grant No.1 202 006).