数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1045-1080.doi: 10.1007/s10473-023-0305-2

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THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS*

Lili WEN1, Engui FAN2, Yong CHEN3,†   

  1. 1. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, and Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200241, China;
    2. School of Mathematical Sciences and Key Laboratory for Nonlinear Science, Fudan University, Shanghai 200433, China;
    3. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, and Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200241, China; College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
  • 收稿日期:2021-09-22 修回日期:2022-06-02 出版日期:2023-06-25 发布日期:2023-06-06
  • 通讯作者: Yong CHEN, E-mail: ychen@sei.ecnu.edu.cn
  • 作者简介:Lili WEN, E-mail: wenllerin@163.com;Engui FAN, E-mail: faneg@fudan.edu.cn
  • 基金资助:
    National Natural Science Foundation of China(12175069 and 12235007) and the Science and Technology Commission of Shanghai Municipality (21JC1402500 and 22DZ2229014).

THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS*

Lili WEN1, Engui FAN2, Yong CHEN3,†   

  1. 1. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, and Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200241, China;
    2. School of Mathematical Sciences and Key Laboratory for Nonlinear Science, Fudan University, Shanghai 200433, China;
    3. School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, and Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200241, China; College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
  • Received:2021-09-22 Revised:2022-06-02 Online:2023-06-25 Published:2023-06-06
  • Contact: Yong CHEN, E-mail: ychen@sei.ecnu.edu.cn
  • About author:Lili WEN, E-mail: wenllerin@163.com;Engui FAN, E-mail: faneg@fudan.edu.cn
  • Supported by:
    National Natural Science Foundation of China(12175069 and 12235007) and the Science and Technology Commission of Shanghai Municipality (21JC1402500 and 22DZ2229014).

摘要: We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with $3\times 3$ matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter $k$ into a single-valued parameter $z$. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the $z$-complex plane is divided into four analytic regions of $D_j: \ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in $D_{j}$, but in the upper-half or lower-half plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in $D_{j}$. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries; these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this $N$-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the $N$-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.

关键词: Sasa-Satsuma equation, nonzero boundary condition, auxiliary eigenfunctions, Riemann-Hilbert problem, soliton solution

Abstract: We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with $3\times 3$ matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter $k$ into a single-valued parameter $z$. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the $z$-complex plane is divided into four analytic regions of $D_j: \ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in $D_{j}$, but in the upper-half or lower-half plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in $D_{j}$. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries; these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this $N$-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the $N$-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.

Key words: Sasa-Satsuma equation, nonzero boundary condition, auxiliary eigenfunctions, Riemann-Hilbert problem, soliton solution