Prolongation Structures and N-Soliton Solutions for a New Nonlinear Schrödinger-Type Equation via Riemann-Hilbert Approach

In this paper, a new integrable nonlinear Schrödinger-type (NLST) equation is investigated by prolongation structures theory and Riemann-Hilbert (R-H) approach. Via prolongation structures theory, the Lax pair of the NLST equation, a 2×2 matrix spectral problem, is derived. Depending on the analysis of red the spectral problem, a R-H problem of the NLST equation is formulated. Furthermore, through a specific R-H problem with the vanishing scattering coefficient, N-soliton solutions of the NLST equation are expressed explicitly. Moreover, a few key differences are presented, which exist in the implementation of the inverse scattering transform for NLST equation and cubic nonlinear Schrödinger (NLS) equation. Finally, the dynamic behaviors of soliton solutions are shown by selecting appropriate spectral parameter λ, respectively.

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Triple-pole soliton solutions of the derivative nonlinear Schrödinger equation via inverse scattering transform

Applied Mathematics Letters ◽

10.1016/j.aml.2021.107741 ◽

2021 ◽

pp. 107741

Author(s):

Nan Liu ◽

Zuxing Xuan ◽

Jinyi Sun

Keyword(s):

Schrödinger Equation ◽

Inverse Scattering ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger ◽

Scattering Transform ◽

Derivative Nonlinear Schrödinger Equation

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The Eigenfunctions and Exact Solutions of Discrete mKdV Hierarchy with Self-Consistent Sources via the Inverse Scattering Transform

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2013.m450 ◽

2015 ◽

Vol 7 (5) ◽

pp. 663-674 ◽

Cited By ~ 1

Author(s):

Q. Li ◽

J. B. Zhang ◽

D. Y. Chen

Keyword(s):

Exact Solutions ◽

Spectral Problem ◽

Inverse Scattering ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

The Self ◽

Scattering Transform ◽

Self Consistent ◽

Exact Soliton Solutions ◽

Mkdv Hierarchy

AbstractAnother form of the discrete mKdV hierarchy with self-consistent sources is given in the paper. The self-consistent sources is presented only by the eigenfunctions corresponding to the reduction of the Ablowitz-Ladik spectral problem. The exact soliton solutions are also derived by the inverse scattering transform.

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Stability of multi-solitons in the cubic NLS equation

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891614500106 ◽

2014 ◽

Vol 11 (02) ◽

pp. 329-353 ◽

Cited By ~ 7

Author(s):

Andres Contreras ◽

Dmitry Pelinovsky

Keyword(s):

Initial Data ◽

Inverse Scattering ◽

Orbital Stability ◽

Inverse Scattering Transform ◽

Nls Equation ◽

Transform Methods ◽

Nonlinear Schrödinger ◽

Scattering Transform ◽

The Stability

We address the stability of multi-solitons for the cubic nonlinear Schrödinger (NLS) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we establish the orbital stability of multi-solitons in the L2(ℝ) space when the initial data is in a weighted L2(ℝ) space.

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Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with non-zero boundary conditions

Physica D Nonlinear Phenomena ◽

10.1016/j.physd.2017.12.007 ◽

2018 ◽

Vol 368 ◽

pp. 22-49 ◽

Cited By ~ 15

Author(s):

Barbara Prinari ◽

Francesco Demontis ◽

Sitai Li ◽

Theodoros P. Horikis

Keyword(s):

Boundary Conditions ◽

Inverse Scattering ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

Nonlinear Schrödinger Equations ◽

Schrödinger Equations ◽

Nonlinear Schrödinger ◽

Scattering Transform ◽

Nonlinear Schrodinger Equations ◽

Square Matrix

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A coupled nonlinear Schrödinger-type equation and its Darboux transformation

Modern Physics Letters B ◽

10.1142/s0217984918501920 ◽

2018 ◽

Vol 32 (17) ◽

pp. 1850192 ◽

Cited By ~ 1

Author(s):

Xianguo Geng ◽

Jiao Wei ◽

Bo Xue

Keyword(s):

Spectral Problem ◽

Trivial Solution ◽

Power Flow ◽

Soliton Solutions ◽

Type Equation ◽

Reduction Technique ◽

Explicit Solutions ◽

Darboux Transformations ◽

Negative Power ◽

Nonlinear Schrödinger

A new coupled nonlinear Schrödinger (NLS)-type equation is proposed by means of the negative power flow of a spectral problem. Resorting to the gauge transformation of the spectral problem and the reduction technique, Darboux transformations for the coupled NLS-type equation and its reduction are constructed, by which explicit solutions of the two coupled NLS-type equations can be engendered from their known solutions. This process can be done continually and will usually yield a series of solutions including multi-soliton solutions. As an illustrate example, one- and two-soliton solutions of the latter coupled NLS-type equation are obtained from a trivial solution.

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