Yindong Zhuang,Yi Zhang,Heyan Zhang and Pei Xia

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

Abstract The purpose of the paper is to formulate multi-soliton solutions for the nonlocal Hirota equations via the Riemann—Hilbert(RH)approach.The RH problems are constructed and the zero structures are studied via performing spectral analysis of the Lax pair.Then we consider three types of nonlocal Hirota equations by discussing different symmetry reductions of the potential matrix.On the basis of the resulting matrix RH problem under the restriction of the reflectionless case,we successfully obtain the multi-soliton solutions of the nonlocal Hirota equations.

Keywords:the nonlocal Hirota equation,Riemann—Hilbert approach,multi-soliton solutions

1.Introduction

As is well known,lots of efficient methods have been available for seeking soliton solutions,including inverse scattering transform[1,2],the RH approach[3—7],Darboux transform[8—10],B¨cklund transform[11]and Hirota bilinear method[12].Among the various methods,the RH approach has proved very powerful in deriving soliton solutions for many completely integrable equations.In fact,the RH approach is a special nonlinear mapping between the group of potentials and the relevant spectral dates.In recent years,a variety of excellent works have been achieved by RH approach,such as generalized Sasa-Satsuma equation[13],matrix modified Korteweg-de Vries equation[14],nonlinear Schr?dinger equation[15]and nonlocal reverse-time nonlinear Schr?dinger equation[16]etc.

At present,we consider the coupled Hirota equations[17]:

where q(x,t)and r(x,t)are both the complex valued functions associated with the real variables x and t.Then a=a1+ia2and b=b1+ib2,a1,a2,b1,b2are real constants.It leads to the Hirota equation[18]which Hirota introduced for the first time in 1973

by the reduction r(x,t)=-q*(x,t),and*is the complex conjugate.When a=1,b=0,equation(2)is reduced to the nonlinear Schr?dinger equation[19],which describes soliton propagation in nonlinear dispersive science.When a=0,b=1,equation(2)is reduced to the modified Korteweg-de Vries equation[20]which is also very representative.As we all know,equation(2)is a well studied nonlinear integrable system and serves as a classical model to describe a variety of nonlinear phenomena in many fields,such as optical fibers,electric communication and engineering fields.Zhang et al investigated the soliton solutions with nonzero boundary conditions[21],and Ankiewicz et al discussed the rogue waves and rational solutions of equation(2)in[22].On the other hand,Guo et al studied the long-time asymptotic behavior of the solution of the Hirota equation by means of the nonlinear steepest descent method[23].Recently,Xu et al obtained the explicit formulas of arbitraryorder multi-pole solutions of the Hirota equation via Darboux transformation and some limit techniques,at the same time,they studied the asymptotic behavior and the soliton interactions of the double- and triple-pole solutions[24].In addition,other important results of the Hirota equation have been obtained in[25—28].

The rest of this paper is as follows.In section 2,we analyze the analyticity of matrix eigenfunctions by introducing the equivalent spectral problem and establish the RH problem on the real-λ line.In section 3,three cases of the nonlocal Hirota equations are discussed,respectively,then the multi-soliton solutions of the nonlocal Hirota equations are derived from a specific RH problem.Finally,a concise summary of this paper will be presented in section 4.

2.The Riemann–Hilbert problem

3.Multi-soliton solutions for the nonlocal Hirota equations

In this section,considering three nonlocal reductions,we solve the RH problem which corresponds to the reflectionless case and obtain multi-soliton solutions of the nonlocal Hirota equations.

3.1.Multi-soliton solutions of reverse-time nonlocal Hirota equation

According to the definition of P+,P-and the scattering relationship between κ+and κ-,it yields

3.2.Multi-soliton solutions of reverse-space nonlocal Hirota equation

3.3.Multi-soliton solutions of reverse-spacetime nonlocal Hirota equation

whereM=(mkj)2N×2Nwith the matrix entries

Next,corresponding to equations(36),(47)and(52)we shall obtain multi-soliton solutions of the nonlocal Hirota equations.When N=1,the soliton solutions for the three types of nonlocal Hirota equations are shown in figure 1,respectively.When N=2,we plot the graphs of the twosoliton solutions for reverse-time nonlocal Hirota equation as shown in figure 2 by selecting appropriate parameters.Due to the different symmetry conditions between the reverse-time nonlocal Hirota equation and the other two nonlocal Hirota equations,we can obtain the different graphs.It is worth noting that when eitherIn other words,λ andis purely imaginary,the two-soliton solutions of the reverse-spacetime nonlocal Hirota equation in figure 1 degenerate into the single-soliton solution in figure 3.

4.Conclusion

The aim of the current investigation is to derive multi-soliton solutions of the nonlocal Hirota equations by utilizing the RH approach.To this end,starting firstly from the equivalence spectral problem of the coupled Hirota equations,which effectively acquire the relevant analytical properties,then we construct the RH problem of the nonlocal Hirota equations.Secondly,considering different symmetry relationships,the three types of nonlocal Hirota equations are discussed.After solving the RH problem without reflection,we finally obtain the multi-soliton solutions of the nonlocal Hirota equations,Particularly,the single- and two-soliton solutions are displayed.

Acknowledgments

This work is supported by the National Natural Science Foundation of China(No.11371326 and No.11975145).