Wen-Xiu Ma

Department of Mathematics,Zhejiang Normal University,Jinhua 321004,China

Department of Mathematics,King Abdulaziz University,Jeddah 21589,Saudi Arabia

Department of Mathematics and Statistics,University of South Florida,Tampa,FL 33620-5700,United States of America

School of Mathematical and Statistical Sciences,North-West University,Mafikeng Campus,Private Bag X2046,Mmabatho 2735,South Africa

Abstarct We conduct two group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems to present a class of novel reduced nonlocal reverse-spacetime integrable modified Korteweg–de Vries equations.One reduction is local,replacing the spectral parameter with its negative and the other is nonlocal,replacing the spectral parameter with itself.Then by taking advantage of distribution of eigenvalues,we generate soliton solutions from the reflectionless Riemann–Hilbert problems,where eigenvalues could equal adjoint eigenvalues.

Keywords:nonlocal integrable equation,soliton solution,Riemann–Hilbert problem

1.Introduction

Group reductions of matrix spectral problems can produce nonlocal integrable equations and keep the corresponding integrable structures that the original integrable equations possess [1–3].If one group reduction is taken,we can obtain three kinds of nonlocal nonlinear Schr?dinger equations and two kinds of nonlocal modified Kortweweg-de Vries(mKdV)equations[1,4].Recently,we have shown that a new kind of nonlocal integrable equations could be generated by conducting two group reductions simultaneously.The inverse scattering transform,Darboux transformation and the Hirota bilinear method can be applied to analysis of soliton solutions to nonlocal integrable equations [5–7].

The Riemann–Hilbert technique has been proved to be another powerful method to solve integrable equations,and especially to construct their soliton solutions [8,9].Various kinds of integrable equations have been investigated via analyzing the associated Riemann–Hilbert problems and we refer the interested readers to the recent studies [10–12]and[3,13–15]for details in the local and nonlocal cases,respectively.In this paper,we would like to present a kind of novel reduced nonlocal integrable mKdV equations by taking two group reductions and construct their soliton solutions through the relectionless Riemann–Hilbert problems.

The rest of this paper is structured as follows.In section 2,we make two group reductions of the Ablowitz–Kaup–Newell–Segur (AKNS) matrix spectral problems to generate type(-λ,λ)reduced nonlocal integrable mKdV equations.Two scalar examples are

and

whereσ=δ=±1.In section 3,based on distribution of eigenvalues,we establish a formulation of solutions to the corresponding reflectionless Riemann–Hilbert problems,where eigenvalues could equal adjoint eigenvalues,and compute soliton solutions to the resulting reduced nonlocal integrable mKdV equations.In the last section,we gives a conclusion,together with a few concluding remarks.

2.Reduced nonlocal integrable mKdV equations

2.1.The matrix AKNS integrable hierarchies revisited

Let us recall the AKNS hierarchies of matrix integrable equations,which will be used in the subsequent analysis.As normal,letλdenote the spectral parameter,and assume thatm,n≥1 are two given integers andp,qare two matrix potentials:

The matrix AKNS spectral problems are defined as follows:

Here the constant square matrices Λ and Ω are defined by

withIsbeing the identity matrix of sizes,andα1,α2andβ1,β2being two arbitrary pairs of distinct real constants.The other two involved square matrices of sizem+nare defined by

called the potential matrix,and

wherea[s],b[s],c[s]andd[s]are defined recursively as follows:

with zero constants of integration being taken.Particularly,we can obtain

and

whereα=α1-α2,β=β1-β2andIm,n=diag (Im,-In).The relations in (6) also imply that

solves the stationary zero curvature equation

which is crucial in defining an integrable hierarchy.

The compatibility conditions of the two matrix spectral problems in (2),i.e.the zero curvature equations

generate one so-called matrix AKNS integrable hierarchy(see,e.g.[16]):

which has a bi-Hamiltonian structure.The second (r=3)nonlinear integrable equations in the hierarchy give us the AKNS matrix mKdV equations:

where the two matrix potentials,pandq,are defined by (1).

2.2.Reduced nonlocal integrable mKdV equations

We would like to construct a kind of novel reduced nonlocal integrable mKdV equations by taking two group reductions for the matrix AKNS spectral problems in(2).One reduction is local while the other is nonlocal (see also [17]for the local case).

Let Σ1,Δ1and Σ2,Δ2be two pairs of constant invertible symmetric matrices of sizesmandn,respectively.We consider two group reductions for the spectral matrixU:

and

where the two constant invertible matrices,Σ andΔ,are defined by

These two group reductions lead equivalently to

and

respectively.More precisely,they enable us to make the reductions for the matrix potentials:

and

respectively.It then follows that to satisfy both group reductions in (12) and (13),an additional constraint is required for the matrix potentialp:

Moreover,we notice that under the group reductions in (12)and (13),we have that

which implies that

and

where s≥0.

Consequently,we see that under the potential reductions(15) and (16),the integrable matrix AKNS equations in (10)withr=2s+1,s≥0,reduce to a hierarchy of nonlocal reverse-spacetime integrable matrix mKdV type equations:

wherepis anm×nmatrix potential which satisfies (19),Σ1,Δ1are a pair of arbitrary invertible symmetric matrices of sizem,and Σ2,Δ2are a pair of arbitrary invertible symmetric matrices of sizen.Each reduced equation in the hierarchy(23) with a fixed integers≥ 0 possesses a Lax pair of the reduced spatial and temporal matrix spectral problems in (2)withr=2s+1,and infinitely many symmetries and conservation laws reduced from those for the integrable matrix AKNS equations in (10) withr=2s+1.

If we fixs=1,i.e.r=3,then the reduced matrix integrable mKdV type equations in (23) give a kind of reduced nonlocal integrable matrix mKdV equations:

wherepis anm×nmatrix potential satisfying (19).

In what follows,we would like to present a few examples of these novel reduced nonlocal integrable matrix mKdV equations,by taking different values form,nand appropriate choices for Σ,Δ.

Let us first considerm=1 andn=2.We take

whereσandδare real constants and satisfyσ2=δ2=1.Then the potential constraint (19) requires

wherep=(p1,p2),and thus,the corresponding potential matrixPreads

Further,the corresponding novel reduced nonlocal integrable mKdV equations become

whereσ=±1.These two equations are quite different from the ones studied in [1,18,19],in which only one nonlocal factor appears.Similarly,if we take

whereσandδare real constants and satisfyσ2=δ2=1 again,then we obtain another pair of novel scalar nonlocal integrable mKdV equations:

whereδ=±1.This pair has a different nonlocality pattern from the one in (28).Moreover,in each of these two equations,there are two nonlocal nonlinear terms,but in each of their counterparts in[1,18,19],there is only one nonlocal nonlinear term.

Let us second considerm=1 andn=4.We take

whereσjandδjare real constants and satisfyσ2j=δ2j=1,j=1,2.Then the potential constraint (19)generates

wherep=(p1,p2,p3,p4),and so the corresponding potential matrixPbecomes

This enables us to obtain a class of two-component reduced nonlocal integrable mKdV equations:

whereσjare real constants and satisfyσ2j=1,j=1,2.

Let us third considerm=2 andn=2.We take

whereσandδare real constants and satisfyσ2=δ2=1.Then the potential constraint (19) tells

and so the corresponding matrix potentials reads

This enables us to get another class of two-component reduced nonlocal integrable mKdV equations:

whereσ=±1.The pattern of the second nonlocal nonlinear terms in these two equations is different from the one in(34).

In the second and third cases,we can also take other similar choices forΣ andΔ as did in the first case,and generate different two-component reduced integrable mKdV equations.

3.Soliton solutions

3.1.Distribution of eigenvalues

Under the group reduction in(12)(or(13)),we can see thatλis an eigenvalue of the matrix spectral problems in (2) if and only if=-λ(or=λ) is an adjoint eigenvalue,i.e.the adjoint matrix spectral problems hold:

wherer=2s+1,s≥0.Consequently,we can assume to have eigenvaluesλ:μ,-μ,and adjoint eigenvalues:-μ,μ,whereμ∈ C.

Moreover,under the group reduction in (12) (or (13)),ifφ(λ)is an eigenfunction of the matrix spectral problems in(2)associated with an eigenvalueλ,thenφT(-λ)Σ (orφT(-x,-t,λ)Δ) presents an adjoint eigenfunction associated with the same eigenvalueλ.

3.2.General solutions to reflectionless Riemann–Hilbert problems

We would like to present a formulation of solutions to the corresponding reflectionless Riemann–Hilbert problems.

LetN1,N2≥0be two integers such thatN=2N1+N2≥1.First,we takeNeigenvaluesλkandNadjoint eigenvaluesas follows:

and

whereμk∈ C,1 ≤k≤N1,andνk∈ C,1 ≤k≤N2,and assume that their corresponding eigenfunctions and adjoint eigenfunctions are given by

respectively.We point out that in the current nonlocal case,we do not have the property

and thus,we need generalized solutions to reflectionless Riemann–Hilbert problems.Such solutions are provided by

whereMis a square matrixM=(mkl)N×Nwith its entries defined by

As shown in[14],these two matricesG+(λ)andG-(λ)solve the reflectionless Riemann–Hilbert problem:

when the orthogonal condition:

is satisfied.

As a consequence of the matrix spectral problems in (2)with zero potentials,we can derive

and based on the preceding analysis,we can take

wherewk,1 ≤k≤N,are constant column vectors.In this way,the orthogonal condition (46) becomes

where 1 ≤k,l≤N.

Now,making an asymptotic expansion

asλ→∞,we obtain

and further,substituting this into the matrix spatial spectral problems,we obtain

This give rise to theN-soliton solutions to the matrix AKNS equation (13):

Here for each 1 ≤k≤N,we have made the splittings,whereandare column and row vectors of dimensionm,respectively,whilevk2and ?vk2are column and row vectors of of dimensionn,respectively.

To presentN-soliton solutions for the reduced nonlocal integrable mKdV equation (23),we need to check ifG+1defined by (51) satisfies the involution properties:

These mean that the resulting potential matrixPgiven by(52)will satisfy the two group reduction conditions in (15) and(16).Therefore,the aboveN-soliton solutions to the matrix AKNS equation (10) reduce to the following class ofN-soliton solutions:

to the reduced nonlocal integrable mKdV equation (23).

3.3.Realization

Let us now check how to realize the involution properties in (54).

First,following the preceding analysis in section 3.1,all adjoint eigenfunctions,1 ≤k≤2N1,can be determined by

and

These choices in (56) (or (57)) engender the selections onwk,1 ≤k≤N:

We emphasize that all these selections aim to satisfy the reduction conditions in (15) and (16).

Now,note that when the solutions to the reflectionless Riemann–Hilbert problems,defined by(43)and(44),possess the involution properties in (54),the corresponding relevant matrixG+1will satisfy the involution properties in(54),which are consequences of the group reductions in (12) and (13).Therefore,when the selections in (58) are made and the orthogonal condition forwkin (49) is satisfied,the formula(55),together with (43),(44),(47) and (48),gives rise toN-soliton solutions to the reduced nonlocal matrix integrable mKdV equation (23).

Finally,let us consider the case ofm=n/ 2=s=N=1.We takeλ1=ν,=-ν,ν∈C,and choose

wherew1,1,w1,2,w1,3are arbitrary complex numbers andSuch a situation leads to a class of one-soliton solutions to the reduced nonlocal integrable mKdV equation(28):

whereν∈Cis arbitrary andw1,1,w1,2∈C are arbitrary but need to satisfywhich is a consequence of the involution properties in (54).

4.Concluding remarks

Type(-λ,λ)reduced nonlocal reverse-spacetime integrable mKdV hierarchies and their soliton solutions were presented.The analysis is based on two group reductions,one of which is local while the other is nonlocal.The resulting nonlocal integrable mKdV hierarchies are different from the existing ones in the literature.

We remark that it would also be interesting to search for other kinds of reduced nonlocal integrable equations from different kinds of Lax pairs [20],integrable couplings [21]and variable coefficient integrable equations [22].In the pair of the considered two group reductions,we can also take

and

with the shifted potentials,wherex0,x0′,t0,t0′are arbitrary constants(see,e.g.[23]).Another interesting topic is to study dynamical properties of exact solutions,including lump solutions [24],soliton solutions [25–27],rogue wave solutions [28,29],solitonless solutions [30]and algebro-geometric solutions [31,32],from a perspective of Riemann–Hilbert problems.All this will greatly enrich the mathematical theory of nonlocal integrable equations.

Acknowledgments

The work was supported in part by NSFC under the grants 11975145,11972291 and 51771083,the Ministry of Science and Technology of China (G2021016032L),and the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17 KJB 110020).