Reverse Space‐Time Nonlocal Sine‐Gordon/Sinh‐Gordon Equations with Nonzero Boundary Conditions

2018/07/19 by Mark J. Ablowitz, Bao‐Feng Feng, Xu‐Dan Luo +1 · 110 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Boundary value problem #Eigenfunction #Eigenvalues and eigenvectors #Geometry #Integrable system #Inverse problem #Inverse scattering problem #Inverse scattering transform #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum mechanics #Scattering #Sine #Soliton #sine-Gordon equation

paper · pdf · doi:10.1111/sapm.12222

published in Studies in Applied Mathematics 141(3), 267-307 (Wiley)

openalex publication_date 2018/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Abstract Nonlocal reverse space‐time Sine/Sinh‐Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well‐known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1‐soliton and 2‐soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered.

Citations

• The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
• Integrable Nonlocal Nonlinear Equations

Cited by

• Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
• Riemann-Hilbert construction of N-soliton solutions for the n-component reverse-space-time nonlocal generalized Sasa-Satsuma system
• Nonlocal massive Thirring model and its solutions
• Riemann-Hilbert approach to the inhomogeneous fifth-order nonlinear Schrödinger equation with non-vanishing boundary conditions
• Hirota bilinear forms of the AKNS(N) systems
• N-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithm
• General soliton and (semi-)rational solutions of the partial reverse space y-non-local Mel'nikov equation with non-zero boundary conditions. [europepmc]

Related

• Inverse scattering transform for the nonlocal reverse space-time Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations with nonzero boundary conditions
• Integrable Nonlocal Nonlinear Equations
• Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions
• The Riemann-Hilbert approach for the nonlocal derivative nonlinear Schrödinger equation with nonzero boundary conditions
• Integrable nonlocal nonlinear equations