COUPLED NONLOCAL NONLINEAR SCHR?DINGER EQUATION AND N-SOLITON SOLUTION FORMULA WITH DARBOUX TRANSFORMATION??

Rui Fan,Fajun Yu

(School of Math.and Systematic Sciences,Shenyang Normal University,Shenyang 110034,Liaoning,PR China)

Abstract Ablowitz and Musslimani proposed some new nonlocal nonlinear integrable equations including the nonlocal integrable nonlinear Schr?dinger equation.In this paper,we investigate the Darboux transformation of coupled nonlocal nonlinear Schr?dinger(CNNLS)equation with a spectral problem.Starting from a special Lax pairs,the CNNLS equation is constructed.Then,we obtain the one-,two-and N-soliton solution formulas of the CNNLS equation with N-fold Darboux transformation.Based on the obtained solutions,the propagation and interaction structures of these multi-solitons are shown,the evolution structures of the one-dark and one-bright solitons are exhibited with N=1,and the overtaking elastic interactions among the two-dark and two-bright solitons are considered with N=2.The obtained results are different from those of the solutions of the local nonlinear equations.Some different propagation phenomena can also be produced through manipulating multi-soliton waves.The results in this paper might be helpful for understanding some physical phenomena described in plasmas.

Keywordscoupled nonlocal nonlinear Schr?dinger(CNNLS)equation;Darboux transformation;dark soliton;bright soliton

1 Introduction

The Schr?dinger equation is one of the basic equations of quantum mechanics proposed by physicist Schr?dinger in 1926.In recently,Ablowitz and Musslimani[1,2]proposed some new nonlocal nonlinear integrable equations which include the nonlocal integrable nonlinear Schr?dinger equation,mKdV equation,and so on.According to the relative scale of the relative length of the root beam width and the nonlinear response function of the medium,the nonlocal nonlinear Schr?dinger equation can be divided into four classes[3]including the local class,weakly nonlocal class,general nonlocal class and strongly nonlocal class.The spatial solitons in nonlocal nonlinear media have attracted great interesting[4-13].The research status of nonlocal spatial solitons were summarized and reviewed in[11].

With the further study of the soliton theory,it provides many methods for solving nonlinear partial differential equations,such as the homogeneous balance method[14],bilinear method[15],traveling wave method[15],Darboux transformation(DT)method[16],inverse scattering transform method[15-19].For examples,some discrete rogue-wave solutions with dispersion in parity-time symmetric potential of Ablowitz-Musslimani equation were derived in[20].Some bright,dark and breather wave soliton solutions of the super-integrable hierarchy were presented by Darboux transformation[21].The non-autonomous multi-rogue wave solutions in a spin-1 coupled nonlinear Gross-Pitaevskii equation with varying dispersions,higher nonlinearities,gain/loss and external potentials were investigated in[22].The generalized three-coupled Gross-Pitaevskii equations by means of the DT and Hirota’s method were worked,and several non-autonomous matter-wave solitons including dark-dark-dark and bright-bright-bright shapes were obtained in[23].The nonautonomous discrete vector bright-dark solutions and their controllable behaviors in the coupled Ablowitz-Ladik equation with variable coefficients were considered in[24].The Darboux transformation method with 4×4 spectral problem are applied to study a specific equation and then the explicit solutions of the lattice integrable coupling equation were obtained in[25,26].

The spectral problem stems from a solution of nonlinear partial differential equations,and the new solution was derived by Darboux transformation method[16].The Darboux transformation can get a new solution from a known equation,also some multi-soliton solutions of the nonlinear partial differential equation can be obtained through multiple Darboux transformation[27-32].The coupled nonlinear schr?dinger equation,which describes a nonlinear diffusion regularity of two nonlinear wave propagation in the medium,not only is applied widely in the field of nonlinear optics,but also plays an important role in meteorology.

Wu and He generated the derivative nonlinear Schr?dinger(NLS)equations,whose nonlocal extensions are from Lie algebra splittings and automorphisms in[33].A chain of nonsingular localized-wave solutions was derived for a nonlocal NLS equation with the self-induced parity-time(PT)-symmetric potential through the N-th Darboux transformation by Li and Xu in[34].Some rational soliton solutions were derived for the parity-time-symmetric nonlocal NLS model with the defocusing-type nonlinearity by the generalized Darboux transformation in[35],which includes the first-order solution,dark-antidark,and antidark-dark soliton.Zhang,Qiu,Cheng and He derived a kind of rational solution with two free phase parameters of nonlocal NLS equation,which satisfies the parity-time(PT)symmetry condition in[36].Some new unified two-parameter wave model,connecting integrable local and nonlocal vector NLS equations were presented by Yan in[37].Ma and Zhu introduced the geometry of a nonlocal NLS equation and its discrete version in[38].

In this paper,we investigate the Darboux transformation of a coupled nonlocal nonlinear Schr?dinger(CNNLS)equation.Starting from a special Lax pairs,we construct a nontrivial single soliton solution from the zero solution,and a double soliton solution from a single soliton solution, finally obtain the N-soliton solution formula of the CNNLS equation with N-fold Darboux transformation.In addition,we find that the coupled nonlocal soliton equations have some richer mathematical structures,the propagation and interaction structures than a single equation.

This paper is organized as follows.In Section 2,we construct Darboux transformation for CNNLS equation,and prove the procedure of DT.In Section 3,we apply the Darboux transformation to CNNLS and obtain N-soliton solution formula.The evolutions of the intensity distribution of the soliton solutions are illustrated in form of figures.

2 Darboux transformation for the coupled nonlocal Schr?dinger equation

There are some important results of coupled Schr?dinger(CNLS)equation in the previous works.In this section,we are motivated by the investigation for the general coupled CNLS equation and consider the coupled nonlocal Schr?dinger(CNNLS)equation as follows:

We first demonstrate the integrability of the CNNLS equation(1),then we construct its Darboux transformation.The Lax pairs of CNNLS equation(1)is given as follows

and

where

and r1=ap?(?x,t)+bq?(?x,t),r2=b?p?(?x,t)+cq?(?x,t),σ3P= ?Pσ3,and σ3P2σ3=P2.p(x,t),q(x,t),r1(x,t),r2(x,t)are the “potential”about x and t,a and c are real,and b is complex.We can check that the zero-curvature equation Ut?Vx+[U,V]=0 leads to equation(1)under the condition of compatibility.

Next,we construct the Darboux transformation of CNNLS equation with the Lax pairs of equations(2)and(3),which are satisfied with the 3×3 transformation matrix of φ,and.We consider the isospectral problem of CNNLS equation and recommend a gauge transformation T of the Lax pairs of equations(2)and(3):

with

where λjandshould choose appropriate parameters,here the determinants of coefficients for equation(8)are nonzero.

We give a 3×3 matrix T,which is of the form as follows

where N is a natural number,(m,n=1,2,3,m≥0)are the functions of x and t.By calculations,we can obtain?T as follows

which proves that λj(1≤ j≤ 3N,)are 3N roots of?T.Based on the above conditions,we will prove thatandhave the same forms with U and V,respectively.

in which the transformation formulas between old and new potentials are shown on as follows

The transformations(13)are used to get a Darboux transformation of the spectral problem(6).

Proof Setting T?1=and

we are easy to verify that Bsl(1≤s,l≤3)are 3N-order or 3N+1-order polynomials in λ.

By some simply calculations,λj(1≤ j≤ 3)are the roots of Bsl(1≤ s,l≤ 3).Hence,equation(14)has the following structure

where

Through comparing the coefficients of λ in equation(17),we have the following system

In the following section,we assume that the new matrixhas the same type with U,which means that they have the same structures only by transform p,q,r1,r2of U into,,,of.After detailed calculation,we compare the ranks of λN,and get the objective equations as follows:

From equations(12)and(13),we know that=C(λ).The proof is completed.

Proposition 2 Under the transformation(19),the matrixdefined by(7)has the same form as V,that is,

Proof We assume the new matrixalso have the same form with V.If we obtain some relations between p,q,r1,r2and,,,similar to(13),we can prove that the gauge transformation T turns the Lax pairs U,V into new Lax pairs,with the same types.

It is easy to verify that Esl(1≤s,l≤3)are 3N+1-order or 3N+2-order polynomials in λ.

Through some calculations,λj(1≤ j≤ 3)are the roots of Esl(1≤ s,l≤ 3).Thus,equation(21)has the following structure

where

Through comparing the coefficients of λ in equation(24),we get the objective equations as follows:

In the above section,we assume the new matrixhas the same type with V,which means they have the same structures only by transform p,q,r1,r2of V intoof.From equations(13)and(19),we know that=F(λ).The proof is completed.

3 N-soliton Solution Formula for the Coupled Nonlocal Schr?dinger Equation

In order to obtain the N-soliton solution formula of CNNLS equation with Darboux transformation,we firstly give a set of seed solutions p=q=r1=r2=0 and substitute the solutions into equations(2)and(3),which can get three basic solutions of CCNLS equation:

Substituting equation(26)into equation(9),we obtain

In order to obtain the one-soliton solution of equation(1).We consider N=1 in equations(10)and(11),and obtain the matrix T

and

Based on equations(9)and(27),we can obtain the following systems

where the analytic one-soliton solutions of CNNLS equation are obtained by the Darboux transformation method as follows

To illustrate the wave propagations of the obtained one-soliton solutions(32),we can choose these free parameters in the forms λ1,λ2,λ3,(m=1,2,3,k=1,2,3)and the intensity distributions for the soliton solutions given by equation(32)are illustrated in Figure 1.From the single soliton,we can find that the amplitude of the bright-dark soliton grows and decays with time depending on the parameters λ1,λ2,λ3,(m=1,2,3,k=1,2,3).

In equation(32),when λ1= λ2= λ3orin the first case,the denominator equation(30)is zero,also?is zero,so there is no solution.In the second case,when the values λ1,λ2,λ3are not exactly same and the valuesare not exactly same,then the equation has some solutions.

Now we consider N=2 in equations(10)and(11),and obtain the following system

where i=0,1,2 and j=1,2,···,6.According to equation(33)and Cramer’s rule,we can obtain

where

Figure 1:(a)The intensity distributionof equation(32)with λ1=0.2,λ2=0.3,λ3=0.5,=0.1+0.2i,=0.6i,=0.5+0.3i,=0.3+0.1i,=0.2+0.3i,=0.2+0.1i;(b)the intensity distributionof equation(32)with λ1=1,λ2=i,λ3=2,=i,=3,=2,=1,=2,=1;(c)the intensity distributionof equation(32)with λ1=0.2, λ2=0.3, λ3=0.1,=1.1+0.2i,=1.6i,=0.5+1.3i,=1.3+0.1i,=1.2+0.3i,=0.2+1.1i;(d)the intensity distribution|of equation(32)with λ1=0.1,λ2=0.2,λ3=0.3,=0.3+0.1i,=0.2+0.3i,=0.3+0.1i,=0.1+0.2i,=0.2+0.4i,=0.5+0.1i.

Based on equations(9)and(27),we can obtain the following systems

where the analytic two-soliton solutions of CNNLS equation are obtained by the DT method as follows

It is shown that solitary waves in nonautonomous nonlinear and dispersive systems can propagate in the form of so-called nonautonomous solitons or soliton-like similaritons.In Figure 2,the amplitude of the bright soliton also grows and decays with time.But the velocities before and after the peak time are different,which can be observed clearly from the nonsymmetric contour plot.The collapsing process after the largest amplitude is quicker,and vanishes rapidly.For illustration,the propagations and evolutions of| ep[2]|,| eq[2]|are shown in Figure 2.

In equation(36),consider three values of λ1,λ2,λ3,λ4,λ5,λ6are zero,or three values are same.In the first case,the numerator denominator is zero.In the second case,if the remaining three values are not identical,then the numerator is zero.Also there is the third case,that is there is one or two zeros,and the front six valuescan not all be zero,with,so that there is a solution.

Figure 2:(a)The intensity distributionof equation(36)with λ1=1,λ2=i,λ3=0,λ4=i,λ5=2i,λ6=0,=0.3i,=0,=0.5i,=0,=0,=0.4i,=0,=0.1i,=0.2i,=0,=0,=2i;(b)the intensity distributionof equation(36)with λ1=1,λ2=i,λ3=2,λ4=i,λ5=2i,λ6=1,=i,=0,=i,=1,=0,=i,=2,=i,=i,=0,=1,=2i;(c)the intensity distributionof equation(36)with λ1=0.1,λ2=i,λ3=0.2,λ4=i,λ5=0.2i,λ6=0.1,=i,=0,=i,=0.1,=0,=i,=0.2,=i,=i,=0,=0.1,=0.2i;(d)the intensity distributionof equation(36)with λ1=0.1,λ2=i,λ3=0.2,λ4=i,λ5=0.2i,λ6=0.1,=i,=0,=i,=0.1,=0,=i,=0.2,=i,=i,=0,=0.1,=0.2i.

In order to obtain the N-soliton solution formula of the CNNLS equation,we consider N=n in equation(10),take i=0,1,2,···,n?1 and j=1,2,···,3n into equation(11).We use the same way to get the form of N-soliton solution formula

and

We can obtain some solution of linear systems via the Cramer’s rule

with

where

Based on equations(9)and(27),we can obtain the following systems

where the analytic N-soliton solutions of CNNLS equation are obtained by the Darboux transformation method as follows

Figures 1 and 2 exhibit the exact one-and two-soliton solutions of equation(1),similar to the one-soliton solution,we obtain the N-soliton solution formula of the CNNLS equation with N-fold Darboux transformation.Based on the obtained solutions,the propagation and interaction structures of these multi-solitons are shown graphically:Figure 1 exhibits the evolution structures of the one-dark and one-dark solitons with N=1;in Figure 2,the overtaking elastic interactions among the twodark and two-bright solitons with N=2.The results in this paper might be helpful for understanding some physical phenomena described in plasmas.

4 Conclusions

In this paper,we give the form of N-soliton solution of CNNLS equation with Darboux transformation.In addition,the one-soliton and two-soliton solutions are solved in detail.At present,few of the CNNLS equations are studied by the Darboux transformation.We obtain the one-,two-and N-soliton solution formulas of the CNNLS equation with N-fold Darboux transformation.Based on the obtained solutions,the propagation and interaction structures of these multi-solitons are shown,the evolution structures of the one-dark and one-bright solitons are exhibited with N=1,and the overtaking elastic interactions among the two-dark and two-bright solitons are considered with N=2.These results might be helpful for understanding physical phenomena in a nonlinear diffusion regularity of two nonlinear wave propagation in the medium,and can be applied widely in the field of nonlinear optics,which also plays an important role in meteorology.