A discrete KdV equation hierarchy:continuous limit,diverse exact solutions and their asymptotic state analysis

Xue-Ke Liu and Xiao-Yong Wen

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

Abstract In this paper,a discrete KdV equation that is related to the famous continuous KdV equation is studied.First,an integrable discrete KdV hierarchy is constructed,from which several new discrete KdV equations are obtained.Second,we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit.Third,the generalized (m,2N-m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair.Finally,the diverse exact solutions including soliton solutions,rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation,and their asymptotic states and physical properties such as amplitude,velocity,phase and energy are analyzed.At the same time,some soliton solutions are numerically simulated to show their dynamic behaviors.The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations.

Keywords:discrete KdV equation hierarchy,continuous limit,generalized (m,2N-m)-fold Darboux transformation,exact solutions,asymptotic analysis

1.Introduction

Recently,the discrete nonlinear differential difference equations,taken as the spatially discrete counterparts of nonlinear partial differential equations,have received widespread attention due to their reach applicability in different physical phenomena [1–14].The motivation of studying discrete equations is that they can maintain certain properties of original corresponding continuous equations,and may be applied in numerical simulation [15].In 1977,the following discrete mKdV lattice equation has been proposed as [16]:

where un=u(n,t) is the function of variables n and t,and in the continuous limit,equation (1)is related to the continuous mKdV equation.Since equation (1) was proposed,some extended discrete mKdV lattice equations have also been presented and studied [17–27].In [18,19],an extended discrete mKdV lattice equation is proposed as

where α is an arbitrary real constant.When α=±1,equation (1) is just equation (2).When α=-1,we only replace t with-t,at this time,the above two equations are completely equivalent.In [17],the author has compared two special cases of equation (2) when α=1 and α=0,and has given their different Lax pairs and solitary wave solutions,and also has found that the first-order solitary wave solutions of the two cases are very different,the solitary wave solutions of the case α=1 are closer to the continuous case which has sech-shaped solitary wave solutions,while the case α=0 is interesting because it has non-vanishing waves with a sech’shape [17].In [18],many explicit rational exact solutions including some new solitary wave solutions of equation (2)have been obtained by using a new generalized ans?tz method.In [19,20],the periodic solutions and solitary solutions of equation (2) have been gained by the modified Jacobian elliptic function method.In [21,22],the new solitary wave solutions have been derived by the hyperbolic function method.In [23],the rational formal solutions have been investigated by the rational expansion method.In[24,25],when α ≠0,equation (2) has been mapped to the famous mKdV and KdV equations by the continuous conditions

and

respectively,and moreover,the Lax pair of equation (2) and its corresponding Lax-integrable coupled extension,which is corresponding to the coupled KdV and mKdV equations by using the continuous limit,have been proposed,and some periodic and solitary wave solutions have been constructed via the function-expansion method.However,when α=0,equations (2) and (5) are equivalent by just replacing t with-t,and equation (5) can not correspond to the continuous mKdV equation under the transformation (3),but it still corresponds to the continuous KdV equation under the transformation (4).In [26],an exact two-soliton solution of equation (2) has been derived by using the Hirota direct approach when α is an arbitrary real constant,and the completely elastic interaction between two solitons has been discussed.In[27],the one-fold Darboux transformation(DT)and explicit solutions of equation (2) have been obtained by one author of this paper when α is an arbitrary real constant,and when α ≠1,its DT is difficult to be extended to higher levels,so that it is difficult for us to obtain its higher-order soliton solution.

Based on previous studies,we know that there has been considerable work done on considering equation (2) with the case α=1 [18–25]and the case α ≠0 [26,27].From the known results in [17],we can clearly see that the two cases α=1 and α=0 of equation (2) are very different.However,little work has been done to apply the generalized (m,2N-m)-fold DT technique to give various exact solutions of equation (2) with α=0.Therefore,it is necessary to do further research on this case,in what follows,we will discuss equation (2) with α=0 [17],that is

which is a little different from the equation in the form in[17],in fact,if we replace unwith-(1+un),equation (5) is the second equation in [17],for the convenience of our later discussion,we have changed the second equation in[17]and its corresponding Lax pair,which admits

where λ is the spectral parameter unrelated to t,E is the shift operator satisfying Ef(n,t)=f(n+1,t),E-1f(n,t)=f(n-1,t),andφn=(φn,ψn)Tis an eigenfunction vector.With the aid of symbolic computation,we can easily find that equation(5)is equivalent to the compatibility condition of Lax pair (6)and (7).

It should be noted that equation(2)can be continuous to the mKdV and KdV equations through the continuous limit when α is a non-zero real constant [24,25].However,when α=0,equation (2) is just equation (5),which can only be continuous to the KdV equation by using the continuous limit,not related to the continuous mKdV equation.Therefore,it is more appropriate for us to describe equation(5)as a discrete KdV equation when α=0,hereafter referred to as the discrete KdV equation.And later,by solving equation(5),we will find that the properties of soliton and rational solutions of equation (5) are very similar to the continuous KdV equation,so that is the main reason why we call it the discrete KdV equation.The discrete generalized(m,2N-m)-fold DT[28,29]or generalized (m,N-m)-fold DT [30]is an effective method to solve the discrete integrable equation,and the main advantage of this method is that it can not only give soliton solutions and rational solutions but also give a variety of mixed solutions.According to the information we have,the relevant hierarchy and generalized (m,2N-m)-fold DT of equation (5) have not been studied,and its soliton,rational and hybrid solutions,and relevant asymptotic state analysis have not been considered before.

Therefore in this work,we will further study equation(5)and give some new results and physical phenomena.The article is organized as follows.In section 2,the hierarchy related to equation (5) will be constructed via Tu scheme technique[5].In section 3,we will study the continuous limit of two new higher-order discrete equations in this hierarchy.In section 4,the discrete generalized (m,2N-m)-fold DT will be constructed.In section 5,the soliton solutions,rational solutions,and their mixed solution of equation (5) are obtained via the resulting DT.Meanwhile,we will also discuss their physical and asymptotic properties through asymptotic analysis.The last section is a brief summary.

2.A discrete hierarchy related to equation (5)

In this section,we will construct the discrete hierarchy related to equation (5) from the discrete matrix spectral problem (6)via Tu scheme technique[5].First,starting from(7),we take

By seeking the solution of the stationary zero curvature equation Pn+1Un-UnPn=0,we have

whereAn(j),Bn(j)andCn(j)are functions related to un.Next,we takeAn(0)=then the rest ofAn(j),Bn(j),Cn(j)will be gained by recursive relation(8).The first few expressions of them are written as

Then by using the matrix Pnwe define a new matrixPn(m)as

According to the relation (8),we obtain

In order to gain the discrete KdV hierarchy we want,we need to find a matrix to modifyPn(m),so a specially modified matrix can be defined as

andVn(m)is defined asVn(m)=Pn(m)+Δn(m).Then we have

Now the time evolution of φncan be expressed asφn,tm=Vn(m)φn,and it meets the compatibility condition

with equation (6),from which we can obtain the following discrete equation hierarchy as

(I) When m=1,equation (9) reduces to

which is the same as equation (5) actually,except that the time parts of their Lax pairs are different,and the new time part of its Lax pair is

However,in fact,equations (7) and (11) can be

equivalent under gauge transformation.(II) When m=2,equation (9) reduces to

and the time part of its Lax pair is

with

(III) When m=3,equation (9) reduces to

and the time part of its Lax pair is

with

When we continue to take the value of m,we will get a series of new discrete equations.These equations may also have some good properties worthy of our further study,such as integrability,Hamiltonian structure and conservation law as done in [5,28,29],which are not our main goal in this paper.Our next main task is to study various exact solutions of equation (5) via the discrete generalized (m,2N-m)-fold DT[28,29]proposed by one of the authors in this paper and analyze their physical properties.Before doing this,we first use the continuous limit technique to correspond the first three equations of this hierarchy to the continuous KdV equation with physical significance.

3.Continuous limit

As we all know,the continuous limit of discrete systems is one of the remarkably important research areas in soliton theory[9–12,24,25].Since the continuity limit of equation (5),as a special case of equation (2),has been discussed [24,25],next we mainly discuss the continuous limits of equations (12) and(14).Using the same approach,equation (12) is mapped into

under the transformation

which happens to be the famous KdV equation if O(δ5) is ignored and τ is changed into t.Here δ is an arbitrary small parameter.

When the function unmeets the following condition

Equation (14) is also transformed into (16).If we neglect its O(δ5),equation (14) is also continuous to the KdV equation.Unfortunately,we can not correspond the higher-order discrete equations in this hierarchy to the higher-order KdV equation.Since equations (12) and (14) also continuously correspond to the KdV equation,we also call them the discrete KdV equations.

Through the above process,we find that the first few discrete equations in this hierarchy can continuously correspond to the continuous KdV equation,so it is appropriate to call the hierarchy in section 2 the discrete KdV hierarchy.The discrete equations in this hierarchy only can correspond to the continuous KdV equation,not correspond to the continuous mKdV equation,although equation (5) is a special case of discrete mKdV equation(2),in fact,it only can correspond to the continuous KdV equation.Therefore,that is another reason why we call this hierarchy the discrete KdV hierarchy.Continuing this process,we can get more discrete KdV equations from the above hierarchy.Although they are all discrete KdV equations,we know that equations (12) and (14) have more nonlinear terms than equation(5),so it is possible that these higher-order equations are more accurate and effective as discrete numerical schemes of continuous KdV equations in some aspects.Whether these equations have new properties is worth further discussion.As far as we know,when α ≠0,equation(2)has been investigated via DT[27],and when α=0,the DT of equation(2)can not be reduced to the DT of equation(5),so when α=0,the special equation(5)need further investigation,which is also our main task below.

4.Discrete generalized (m,2N-m)-fold DT

As mentioned above,equation(5)is a special case of(2)when α=0.For this special case,its generalized (m,2N-m)-fold DT has not been studied,and the generalized(m,2N-m)-fold DT of newly obtained equations (12) and (14) have not been studied.Therefore,in this section,we take equation (5) as an example to discuss its generalized (m,2N-m)-fold DT through the Taylor expansion and a limit procedure [28,29].First,we consider a gauge transformation as

which can yield the following equations

where N is a positive integer,andan(0),an(2j-2),bn(2j-1),cn(2j-1)anddn(2j)(j=1,2,...,N)are 4N functions of the variables n and t which are determined later.LetT(λi+ε)=then we have

in which ε is an arbitrary small parameter.In order to find the solutions of the 4N unknowns of Tnwhen m(m ＜2N) spectral parameters λi(i=1,2,…,m) are taken,we only need to solve the following 4N equations

where nonnegative integers m,miand N satisfy the relation 2N=m+∑im=1mi.According to the above analysis,we can give the following discrete generalized (m,2N-m)-fold DT theorem:

Theorem 1.Letφn(λi)=(φn(λi),ψn(λi))Tbe m column vector solutions of Lax pair (6) and (7) with the spectral parametersλi(i=1,2,…,m),and unis the initial solution of equation (5),then the generalized(m,2N-m)-fold DT about the old solution unand the new oneis given by

with

where Δ1=det([Δ1(1),Δ1(2),...,Δ1(m)]T),Δ2=det([Δ2(1),1≤i≤m)are given by following formulae:and Δan(0)is derived by replacing the Nth column of determinant Δ1with(1 ≤j≤mi+1,1≤i≤m).Similarly,andare obtained by replacing first and(N+1) th column of determinantΔ2withandrespectively.

Remark 1.We need to note that m represents the number of spectral parameters and(2N-m)represents the number of equations that we need to get from (22).Whenm=2Nandmi=0,the generalized(m,2N-m)-fold DT becomes the usual 2N-fold DT,which can get the soliton solutions;when m=1 andmi=2N-1,theorem 1 reduces to(1,2N-1)-fold DT which can get the rational solutions;If m=2,theorem 1 reduces to(2,2N-2)-fold DT,from which we can obtain the mixed solutions.For cases other than those discussed above,we can obtain more kinds of mixed interaction solutions.

5.Diverse exact solutions and their asymptotic analysis

Asymptotic analysis is a very important method in integrable systems,which can be used to study physical phenomena and asymptotic behaviors.Recently,the asymptotic behaviors of soliton solutions and rational solutions of some special equations have been obtained by an improved asymptotic analysis method which relies on the balance between different algebraic terms up to the subdominant level [31–33],such as the Gerdjikov-Ivanov equation[31],Hirota equation[32]and defocusing nonlocal nonlinearSchr?dinger equation [33].In this part,we will use the generalized (m,2N-m)-fold DT to give diverse exact solutions on non-zero seed background of equation (5),and then discuss their asymptotic behaviors by making an asymptotic analysis technique.

5.1.Soliton solutions and their physical properties and asymptotic analysis

Let us rewrite the solution of Lax pair(6)and(7)as φn=(φn,ψn),in whichφn=(φ1,n,φ2,n)T,ψn=(ψ1,n,ψ2,n)T.Taking the trivial seed solution un=1 and substituting it into the Lax pair (6) and (7),its basic solution are given by

with

where|λ|＞2.Furthermore,we define(i=1,2,…,N),where ri(i=1,2,…,N) are arbitrary constants,then we can obtain multi-soliton solutions of equations (5) from (23) by the usual 2N-fold DT.In order to understand them more vividly,we draw their structures when N=1 as shown in figure 1.

When N=1,λ=λi(i=1,2),we can obtain the following exact solutions as

(I) If one of the parameters λ1and λ2is equal to 2,without losing generality,let λ2=2,we can get the one-soliton solution by the expression (26).For analyzing its properties,we rewrite it as

with

where |λ1|＞2.From (27),we can analyze its physical properties such as amplitude,wave number,wave width,velocity,phase and energy,which are listed in table 1,in which the energy ofis calculated byFigures 1(a1) and (b1) show the bell-shaped structure of a one-soliton solution with parameters λ1=4,λ2=2,from which we can clearly see that one soliton maintains its shape and velocity during the propagation.

In order to analyze the propagation stability of the one-soliton solution,we numerically simulate it by using the finite difference method [34],which is given by figure 2.Figure 2(a) shows the exact one-soliton solution,which is consistent with figure 1(a1).Figure 2(b) is the numerical solution without any noise,which is almost the same as the exact solution in figure 2(a).Figure 2(c) is the time evolution of the numerical solution after increasing a 2%noise to an exact solution.We can find that it has little change compared with the numerical solution in figure 2(b).When we increase a 5% noise,the numerical solution will change more obviously,but its overall shape will not change greatly,as is shown in figure 2(d).The numerical simulation result further illustrates the correctness of the one-soliton solution,and the soliton propagation resists small noise and keeps stability in a relatively short time.

Figure 1.(a1) Bell-shaped one-soliton structure with parameters λ1=4,λ2=2,r1=-r2=1;(b1) The propagation processes for onesoliton solution at t=-2(dashdot line),t=0(longdash line)and t=2 (solid line).(a2)Bell-shaped two-soliton structure with parameters λ1=,λ2=3,r1=-r2=-1;(b2)The propagation processes for two-soliton solution at t=-6(dashdot line),t=0(longdash line)and t=6 (solid line).

Figure 2.One-soliton solution(27)with the same parameters as in figure 1(a1).(a)Exact solution;(b)numerical solution without any noise;(c) numerical solution with a 2% noise;(d) numerical solution with a 5% noise.

Table 1.Physical characteristics of one-soliton solution.

(II) If neither λ1nor λ2are equal to 2,we can derive the two-soliton solution from (26).In the same way,the two-soliton exact solution can be written as

Before the interaction (t→-∞):

in whichu1-,u2-stand for the limit states of equation (28)before they interact.

After the interaction (t→+∞):

whereu1+,u2+stand for the limit states of equation (28) after they interact.

From the above analysis,we can expose the physical properties of a two-soliton solution (28),which are listed in table 2,from which we know that the amplitude,velocity and energy ofdo not change before and after the interactions,but its phases have changed,so the collision interaction of two solitons in (28) is elastic.

Using the same method as the numerical simulation of one-soliton,we numerically simulate the two-soliton solution(28).The relevant evolution result is shown in figure 3,figure 3(a) represents the exact two-soliton solution,figure 3(b) shows the numerical evolution of the two-soliton solution without any noise.Figures 3(c) and (d) exhibit the numerical evolutions by adding 2% and 5% noises,respectively.From figure 3,we know that the numerical evolution of the two-soliton solution is robust against a small noise in a relatively short time.

Figure 3.Two-soliton solution(28)with the same parameters as in figure 1(a2).(a)Exact solution;(b)numerical solution without any noise;(c) numerical solution with a 2% noise;(d) numerical solution with a 5% noise.

5.2.Rational solutions and asymptotic analysis

In this subsection,we will give the rational solutions of equation(5)with the initial seed solution un=1 by using the generalized (1,2N-1)-fold DT.For the convenience of study,we rewrite solution (25) as

with

in which

Table 2.Physical characteristics of the two-soliton solution.

where ξ=n+2t,and the rest of(j=3,4,5,…) are left out here.Next,taking N=1,2,we give the rational solutions of equation (5).

(I) When N=1,we can get the first-order rational solution by the generalized (1,1)-fold DT as follows:

Table 3.Main mathematical features of rational solution of order N.

Table 3.Main mathematical features of rational solution of order N.

Highest powers in the denominator 1 1 2 2 2 1 12 12 N Background of un Highest powers in the numerator 30 30............N 1 3 1 2N(2N-1) 2N(2N-1)

from which we can see that solution (35) has a singularity at the straight line ξ+2e0=0.Since e0is an arbitrary real parameter,we can move solution (35)with the singular straight line to any position by changing the value of e0.What’s more,→1when n→±∞or t→±∞.And we find that the wave width,wave number,velocity and phase of solution(35)are 1,1,-2,2e0,respectively.It should be pointed out that the primary phase depends on the value of e0,which can control the position of the first-order rational solution (35).

(II) When N=2,we can get the second-order rational solution by the generalized (1,3)-fold DT as

with

where e0,e1are arbitrary real constants.In order to understand the properties of solution (37),we can get its limit states with the aid of symbolic computation Maple.If takingit will yield four kinds of asymptotic states as the following shows.

(i) Ifζ1=n+2t-is regarded as a fixed constant,we can obtainand ζ2→±∞when t→±∞,from which we can get that the first two limit states are

(ii) Ifζ2=n+2t-is regarded as a fixed constant,we can obtain and ζ1→?∞when t→±∞,from which we can get that the last two limit states are

From the above analysis,we can find that two curves ζ1+2e0=0 and ζ2+2e0=0 are the central trajectories of solution,which are also its two singularity lines.The limit states of the second-order rational solution(37)are similar to the first-order rational solution,and the difference is that the first-order rational solution’s singularity line is a straight line,while the second-order rational solution’s singularity line is a curve.Similarly,we can change the position of the singularity line by the constant e0.Using the same method as above,we can analyze the asymptotic behaviors of more complex higher-order rational solutions,which are omitted to discuss here.Next we only give some simple mathematical features of rational solutions of equation (5) listed in table 3,which also reflects their good mathematical laws.

In table 3,the first column represents the order number of rational solution,the second column is its initial background,and the third and fourth columns are the highest powers of the numerator and denominator of it respectively.With the increase of the order number,the background remains unchanged,and the highest powers in the numerator and denominator of the rational solution increase accordingly,but they are always equal.

Remark 2.It should be noted here that for discrete KdV equation (5),its soliton and rational solutions are similar to that of the continuous KdV equation,that is,soliton solutions have no singularity,while the rational solutions have singularity [35].The only difference is that for the discrete KdV equation,its soliton solution is on the non-zero seed background,and for the continuous KdV equation,its soliton solution is on the zero seed background.

5.3.Mixed solution and asymptotic state analysis

In this section,we will discuss the mixed solution of equation(5)by using the generalized(2,2N-2)-fold DT and analyze its asymptotic behaviors.Next,we only discuss the case of N=1.

Different from the previous steps for solving soliton and rational solutions,here we choose two spectral parameters λ1and λ2,and takewith λ1=2.For λ1,we make the Taylor expansion as (32) by expanding φnof (31) at ε=0,while for λ2,we do not make Taylor expansion,then the mixed solution of equation (5) is written as

(i) If ξ1is equal to a fixed constant as t→±∞,we can get ξ2→±∞,and two asymptotic states of solution (41)are

whereu1-,u1+stand for the limit states of solution (41)when t→-∞and t→+∞,respectively,from which we can deduce that the asymptotic expressions are singular at the straight line ξ1=0,and the asymptotic hyperbolic function form solutions do not change its propagation direction before and after the interactions.

(ii) If ξ2is equal to a fixed constant as t→±∞,we can get ξ1→±∞,and two asymptotic states of solution (41)are listed as follows:

Before the interaction (t→-∞):

After the interaction (t→+∞):

whereu2-,u2+stand for the limit states of solution(41)before and after the interactions,respectively.

Remark 3.It should be noted here that for this case,we should get mixed superposition solutions of usual soliton and rational solution in theory.However,from the above analysis,we know that whenξ1is fixed andξ2→±∞,the asymptotic states of the mixed solution(41)become singular at a straight line atξ1=0before and after the interaction,and do not change their propagation direction,while whenξ2is fixed andξ1→±∞,the asymptotic states change their phases.The rational solution propagates along a singular straight linebefore the interaction,while the rational solution propagates along a singular straight lineafter the interaction.Therefore,we only derive mixed superposition solutions of singular soliton and rational solution,and the soliton’s singularity in mixed superposition solution may be caused by the addition of the rational solutions.The mixed solution has three singular straight linesξ1=0,from which we can see the singular straight lines of rational solutions have an arbitrarily real constant e0,so we can control the interaction position of the mixed solution by controlling the value of e0.

6.Conclusions

In this paper,we have studied some properties of a discrete KdV equation (5) and obtained its exact solutions such as soliton solutions,rational solutions,and mixed solutions on a non-zero seed background,which reflects the good physical significance of the discrete KdV equation.The main achievements of this paper are as follows:First,we have constructed the hierarchy of the discrete KdV equation and obtained two new discrete KdV equations,(12) and (14).Second,we have mapped the discrete equations (12) and (14) to the KdV equation by using the continuous limit.Third,we have constructed the discrete generalized(m,2N-m)-fold DT of equation (5) in theorem 1,which is different from the usual N-fold DT,to give soliton,rational and mixed solutions.The structures of one-soliton and two-soliton solutions are demonstrated in figure 1,and their physical properties are listed in tables 1 and 2,respectively.Moreover,we also numerically simulate one-soliton and two-soliton solutions in order to better understand their dynamic behaviors as shown in figures 2 and 3.At the same time,the asymptotic states of various exact solutions have also been analyzed.These results given in this paper might help us understand some physical phenomena described by KdV equations.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No.12071042) and the Beijing Natural Science Foundation (Grant No.1202006).

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