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Higher order nonlinear Schrödinger equation in domains with moving boundaries

Raul Nina Mollisaca, Mauricio Sepúlveda Cortés, Rodfigo Véjar Asém, Octavio Vera Villagrán

TL;DR

The paper addresses the IBVP for a higher-order nonlinear Schrödinger equation in a moving-boundary domain, formulated as $i v_{\tau} + \gamma v_{\xi\xi} + i\chi v_{\xi\xi\xi} = |v|^{2}v$, and develops a moving-boundary transformation to a fixed cylinder to enable analysis. It establishes global existence, uniqueness, and exponential decay of solutions, and introduces a conservative Crank–Nicolson finite-difference scheme with convergence guarantees, validated by numerical experiments on soliton dynamics under domain variation. The scheme is shown to be L2-stable and convergent with truncation error of order $O(\Delta t + \Delta x^{2})$, and numerical tests illustrate traveling solitons and L2-norm behavior as the domain length changes. Overall, the work advances both the theoretical understanding of dispersive PDEs with moving boundaries and practical numerical methods for simulating such systems in cylindrical domains, with a gauge-transformation appendix to simplify the transformed dynamics.

Abstract

The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for a higher order nonlinear Schrödinger (HNLS) equation is considered. Existence and uniqueness of global weak solutions are proved as well as the stability of the solution. Additionally, a conservative numerical method of finite differences is introduced that also verifies stability properties with respect to the $L^2$-norm, and along with proving its convergence, some interesting numerical examples are shown that illustrate the behavior of the solution.

Higher order nonlinear Schrödinger equation in domains with moving boundaries

TL;DR

The paper addresses the IBVP for a higher-order nonlinear Schrödinger equation in a moving-boundary domain, formulated as , and develops a moving-boundary transformation to a fixed cylinder to enable analysis. It establishes global existence, uniqueness, and exponential decay of solutions, and introduces a conservative Crank–Nicolson finite-difference scheme with convergence guarantees, validated by numerical experiments on soliton dynamics under domain variation. The scheme is shown to be L2-stable and convergent with truncation error of order , and numerical tests illustrate traveling solitons and L2-norm behavior as the domain length changes. Overall, the work advances both the theoretical understanding of dispersive PDEs with moving boundaries and practical numerical methods for simulating such systems in cylindrical domains, with a gauge-transformation appendix to simplify the transformed dynamics.

Abstract

The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for a higher order nonlinear Schrödinger (HNLS) equation is considered. Existence and uniqueness of global weak solutions are proved as well as the stability of the solution. Additionally, a conservative numerical method of finite differences is introduced that also verifies stability properties with respect to the -norm, and along with proving its convergence, some interesting numerical examples are shown that illustrate the behavior of the solution.
Paper Structure (14 sections, 5 theorems, 162 equations, 4 figures, 1 table)

This paper contains 14 sections, 5 theorems, 162 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moving Boundary Transformation
  4. Main Result
  5. Existence of solutions
  6. Uniqueness of solutions
  7. Stability
  8. Numerical Scheme
  9. Notation.
  10. Foundations of the Numerical Scheme.
  11. Properties
  12. Convergence.
  13. Numerical experiments.
  14. Appendix: The Gauge transformation

Key Result

Theorem 3.1

Let $u_0\in H^4(0,1)$, there is one classic solution of eq47-eq50 (or 101).

Figures (4)

  • Figure 1: Left: Energy at $L^2$ level. Right: Energy at $H$ level.
  • Figure 2: Numerical solution.
  • Figure 3: Left: Energy at $L^2$ level. Right: Numerical solution. In both cases, $x_0 = -40$ and $x_f = 40$.
  • Figure 4: Numerical solution for the two-soliton case, with its respective $L^2-$ time evolution plot.

Theorems & Definitions (8)

  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 6.2