Inverse scattering transform of the coupled modified Korteweg-de Vries equation with nonzero boundary conditions

In this work, we extend the Riemann-Hilbert (RH) method in order to study the coupled modified Korteweg-de Vries equation (cmKdV) under nonzero boundary conditions (NZBCs), and successfully find its solutions with their various dynamic propagation behaviors. In the process of spectral analysis, it is necessary to introduce Riemann surface to avoid the discussion of multi-valued functions, and to obtain the analytical and asymptotic properties needed to establish the RH problem. The eigenfunction have a column that is not analytic in a given region, so we introduce the auxiliary eigenfunction and the adjoint matrix, which is necessary to derive the analytical eigenfunctions. The eigenfunctions have three kinds of symmetry, which leads to three kinds of symmetry of the scattering matrix, and the discrete spectrum is also divided into three categories by us. The asymptoticity of the modified eigenfunction is derived. Based on the analysis, the RH problem with four jump matrices in a given area is established, and the relationship between the cmKdV equation and the solution of the RH problem is revealed. The residue condition of reflection coefficient with simple pole is established. According to the classification of discrete spectrum, we discuss the soliton solutions corresponding to three kinds of discrete spectrum classification and their propagation behaviors in detail. Project supported by the Fundamental Research Fund for the Central Universities under the grant No. 2019ZDPY07. Corresponding author. E-mail addresses: sftian@cumt.edu.cn and shoufu2006@126.com (S. F. Tian) Preprint submitted to Journal of LTEX Templates April 7, 2021

[1]  Yang Chen,et al.  The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials , 2019, Studies in Applied Mathematics.

[2]  Shou-Fu Tian,et al.  Initial–boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method☆ , 2017 .

[3]  K. Kozlowski,et al.  Riemann–Hilbert approach to a generalized sine kernel , 2019, Letters in Mathematical Physics.

[4]  Bo Xue,et al.  Explicit solutions and conservation laws of the coupled modified Korteweg–de Vries equation , 2015 .

[5]  Gino Biondini,et al.  Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions , 2014 .

[6]  Xianguo Geng,et al.  Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation , 2017, Commun. Nonlinear Sci. Numer. Simul..

[7]  G. Kovačič,et al.  The focusing Manakov system with nonzero boundary conditions , 2015 .

[8]  Wenxiu Ma,et al.  The Hirota-Satsuma Coupled KdV Equation and a Coupled Ito System Revisited , 2000 .

[9]  Robert M. Miura,et al.  Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation , 1968 .

[10]  Xianguo Geng,et al.  A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations , 1999 .

[11]  Shou-Fu Tian,et al.  Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval , 2018 .

[12]  Gino Biondini,et al.  Inverse Scattering Transform for the Defocusing Manakov System with Nonzero Boundary Conditions , 2015, SIAM J. Math. Anal..

[13]  X. Geng,et al.  Riemann–Hilbert approach and N-soliton solutions for a generalized Sasa–Satsuma equation , 2016 .

[14]  Yang Chen,et al.  Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent , 2017, Journal of Mathematical Analysis and Applications.

[15]  Decio Levi,et al.  A hierarchy of coupled Korteweg-de Vries equations , 1983 .

[16]  Xianguo Geng,et al.  Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy , 2014 .

[17]  Zhenya Yan,et al.  Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions , 2020 .

[18]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[19]  J. Lenells,et al.  Asymptotics for the Sasa–Satsuma equation in terms of a modified Painlevé II transcendent , 2019, Journal of Differential Equations.

[20]  A. Fordy,et al.  Generalised KdV and MKdV equations associated with symmetric spaces , 1987 .

[21]  Q. P. Liu,et al.  New Darboux transformation for Hirota–Satsuma coupled KdV system , 2002 .

[22]  X. Geng,et al.  Long-time asymptotics of the coupled modified Korteweg–de Vries equation , 2019, Journal of Geometry and Physics.

[23]  Shuai‐Xia Xu,et al.  Gap Probability of the Circular Unitary Ensemble with a Fisher–Hartwig Singularity and the Coupled Painlevé V System , 2019, Communications in Mathematical Physics.

[24]  N. Yajima,et al.  A Class of Exactly Solvable Nonlinear Evolution Equations , 1975 .

[25]  Shou-Fu Tian,et al.  Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method , 2017 .

[26]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[27]  Nan Liu,et al.  A Riemann–Hilbert approach for a new type coupled nonlinear Schrödinger equations , 2018 .

[28]  Shou‐Fu Tian,et al.  Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition , 2017 .

[29]  Deng-Shan Wang,et al.  Integrable properties of the general coupled nonlinear Schrödinger equations , 2010 .

[30]  Allan P. Fordy,et al.  On the integrability of a system of coupled KdV equations , 1982 .

[31]  Jonatan Lenells.  Initial-boundary value problems for integrable evolution equations with 3×3 Lax pairs , 2012 .

[32]  G. Biondini,et al.  On the focusing non-linear Schrödinger equation with non-zero boundary conditions and double poles , 2017 .

[33]  B. Guo,et al.  Long-time asymptotics for the Sasa–Satsuma equation via nonlinear steepest descent method , 2018, Journal of Mathematical Physics.

[34]  Wen-Xiu Ma,et al.  Riemann–Hilbert problems andN-soliton solutions for a coupled mKdV system , 2018, Journal of Geometry and Physics.

[35]  E. Fan,et al.  Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation , 2001 .

[36]  Shou‐Fu Tian,et al.  Riemann–Hilbert problem for the modified Landau–Lifshitz equation with nonzero boundary conditions , 2019, Theoretical and Mathematical Physics.

[37]  Hiroaki Ono.  On a Modified Korteweg-de Vries Equation , 1974 .

[38]  闫振亚,et al.  New explicit exact solutions for a generalized Hirota-Satsuma coupled KdV system and a coupled MKdV equation , 2003 .

[39]  Zhenya Yan,et al.  The Derivative Nonlinear Schrödinger Equation with Zero/Nonzero Boundary Conditions: Inverse Scattering Transforms and N-Double-Pole Solutions , 2020, Journal of Nonlinear Science.