Nonlocal Nonlinear Schrödinger Equations and Their Soliton Solutions

Abstract

We study standard and nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function $|q(t,x){|}^2$ for the standard and nonlocal NLS equations.

22 pages, 10 figures(typo corrections)

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Cited by in corpus (26)

• Nonlocal Nonlinear Schrödinger Equations and Their Soliton Solutions
• General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions
• A new nonlocal nonlinear Schroedinger equation and its soliton solutions
• Mixed soliton solutions of the defocusing nonlocal nonlinear Schrodinger equation
• Integrable Kuralay equations: geometry, solutions and generalizations
• Integrable nonlocal Hirota equations
• Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data
• Integrable nonlocal asymptotic reductions of physically significant nonlinear equations
• Solutions to integrable space-time shifted nonlocal equations
• Long-time asymptotics for the integrable nonlocal focusing nonlinear Schrödinger equation for a family of step-like initial data
• General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric system
• Nonlocal KdV Equations
• $(2+1)$-Dimensional Local and Nonlocal Reductions of the Negative AKNS System: Soliton Solutions
• Explicit solutions for nonlocal NLS: GBDT and algebro-geometric approaches
• Rational solutions of the defocusing nonlocal nonlinear Schrodinger equation: Asymptotic analysis and soliton interactions
• Discrete Symmetries and Nonlocal Reductions
• Nonlocal Coupled HI-MKdV Systems
• Nonlocal Hydrodynamic Type of Equations
• The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra
• Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau-Lifschitz equation
• Local and nonlocal $(2+1)$-dimensional Maccari systems and their soliton solutions
• Multi-component AKNS systems
• Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
• Symmetries and reductions of integrable nonlocal partial differential equations
• Soliton solutions of the shifted nonlocal NLS and MKdV equations
• The Method of ${\mathcal{M}}_n$-Extension: The KdV Equation