Yunyun Zhai,Lifei Wei,Xianguo Geng and Jiao Wei

School of Mathematics and Statistics,Zhengzhou University,100 Kexue Road,Zhengzhou,Henan 450001,China

Abstract The AB system is the basic integrable model to describe unstable baroclinic wave packets in geophysical fluids and the propagation of mesoscale gravity flows in nonlinear optics.On the basis of the spectral analysis of a Lax pair and the inverse scattering method,we establish the Riemann–Hilbert problem of the AB system.Then,the inverse problems are formulated and solved with the aid of the Riemann–Hilbert problem,from which the potentials can be reconstructed according to the asymptotic expansion of the sectional analytic function and the related symmetry relations.As an application,we obtain the multi-bright-dark soliton solutions to the AB system in the reflectionless case and discuss the dynamic behavior of elastic soliton collisions by choosing appropriate free parameters.

Keywords: AB system,multi-soliton structures,Riemann–Hilbert problem

1.Introduction

In recent decades,integrable systems have attracted extensive attention in describing nonlinear phenomena in various areas,such as fluid mechanics,nonlinear optics,Bose–Einstein condensates,plasma physics and other fields[1,2].A crucial feature of an integrable system is that it can be expressed as a compatibility condition of two linear spectral problems,i.e.a Lax pair,which enables researchers to investigate it via the inverse scattering method[1],Darboux transformation[3–6],algebra-geometric method[7–9],Riemann–Hilbert approach[10–15],etc.The Riemann–Hilbert approach is an effective tool for solving the integrable models and studying their longtime asymptotics and other properties.

In this paper,we will use the Riemann–Hilbert approach to investigate the followingABsystem

with a normalization condition

whereA=A(x,t)is a complex function andB=B(x,t)is a real function[16].TheABsystem,first proposed by Pedlosky using the singular perturbation theory[17],is a significant integrable model since it can describe unstable baroclinic wave packets in geophysical fluids and the propagation of mesoscale gravity flow in nonlinear optics[18–21].Moreover,theABsystem can be reduced to the sine-Gordon equation for realAor the self-induced transparency equations for complexA[22].Rogue wave solutions,breathers andNsoliton solutions for theABsystem have been derived by resorting to the Darboux transformation[23,24]and the dressing method[25].Multi-dark-dark solitons and multibright-bright solitons have been found for repulsiveABsystem via determinants[26,27].Recently,long-time asymptotics of solutions for theABsystem with initial value problems have been studied through the nonlinear steepest-decent method[28].For ultra-short optical pulse propagation models such as the short pulse equation,the associated Riemann–Hilbert problem and explicit soliton formulae may help to better understand the propagation mechanism and characteristics[29,30].Therefore,the present paper is devoted to exploring theABsystem by utilizing the Riemann–Hilbert approach,from which explicit multi-bright-dark soliton solutions in the reflectionless case are obtained with potential functionsAdecaying to zero andBdecaying to 1 at sufficiently fast rates asx→∞.In particular,we find that the multi-soliton collisions are elastic.

The arrangement of this paper is as follows.In section 2,we construct the sectional analytic function and establish the Riemann–Hilbert problem on the basis of the spectral analysis.With the aid of symmetry relations,we solve the nonregular matrix Riemann–Hilbert problem.Section 3 focuses on the time evolution of scattering data and the reconstruction of potentials.In section 4,we obtain explicitN-soliton solutions to theABsystem in the reflectionless case.In particular,we find that the soliton collisions are elastic.The last section contains some discussions.

2.Riemann–Hilbert problem

3.Reconstruction of potentials

Figure 1.One-soliton solution.

4.Soliton solutions

Figure 2.Two-soliton solution.

Figure 3.Two-soliton elastic collision.

Figure 4.Three-soliton solution.

Figure 5.Three-soliton elastic collision.

5.Conclusion and discussion

It is of great significance to investigate the solutions of theABsystem,which may help to better understand ultra-short optical pulse propagation in nonlinear optics.TheABsystem is the negative flow of the Lax pair,so it is very difficult to deal with the system using the Riemann–Hilbert approach.The potentialBis at the diagonal position of the matrixV,which is different from other models such as focusing nonlinear Schr?dinger equation.Therefore,we need to assume functionBdecays to 1 asx→∞and use evolution equations of the Jost function with respect to bothxandtwhen reconstructing the potentials.For the direct scattering problem,the analyticities,symmetries and asymptotic behaviors of the Jost solutions,scattering matrix and discrete spectra,are established by resorting to the inverse scattering transformation.The inverse problems are formulated and solved with the aid of the matrix Riemann–Hilbert problems,and the reconstruction formulas are obtained.And from that we construct the muti-bright-dark soliton solutions to theABsystem.The collisions are elastic,which means that the solitons are capable of propagating over long distances without shape-changing and thus are quite important in optical fiber communication.We also try to find out whether inelastic(shape-changing)collisions exist in the multi-soliton collisions of theABsystem.In addition,the study of non-zero boundary problems by the Riemann–Hilbert method has aroused great interest.In the future,we will focus on finding some other explicit solutions such as the breathers,rogue wave solutions and others by using the inverse scattering method with nonzero boundary conditions.

Acknowledgments

This work is supported by the National Natural Science Foundation of China(Grant Nos.11971441,11 931017,11871440,11901538)and Key Scientific Research Projects of Colleges and Universities in Henan Province(No.20A110006).