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Solitons of the constrained Schrödinger equations

V. E. Vekslerchik

TL;DR

This work addresses a constrained vector Schrödinger equation, showing that the nonlinear system induced by the quadratic constraint can be embedded into the Ablowitz-Ladik hierarchy to yield $N$-soliton solutions. By constructing a two-field formulation and leveraging ALH tau-functions and Miwa shifts, the authors derive explicit multisoliton solutions and reveal a link to a vector NLSE with gradient-type nonlinearity. The one-soliton example exhibits a bright component and a dark component, illustrating rich soliton structure, while the $N$-soliton construction provides a scalable framework controlled by spectral data and mixing matrices. The results offer strong evidence for integrability of the constrained model and show practical applicability to gradient NLSEs, motivating further study of conservation laws and full inverse scattering formulations.

Abstract

We consider the linear vector Schrödinger equation subjected to quadratic constraints. We demonstrate that the resulting nonlinear system is closely related to the Ablowitz-Ladik hierarchy and use this fact to derive the N-soliton solutions for the discussed model. As an example of application of these results we present solitons of some vector nonlinear Schrödinger equation with gradient nonlinearity.

Solitons of the constrained Schrödinger equations

TL;DR

This work addresses a constrained vector Schrödinger equation, showing that the nonlinear system induced by the quadratic constraint can be embedded into the Ablowitz-Ladik hierarchy to yield -soliton solutions. By constructing a two-field formulation and leveraging ALH tau-functions and Miwa shifts, the authors derive explicit multisoliton solutions and reveal a link to a vector NLSE with gradient-type nonlinearity. The one-soliton example exhibits a bright component and a dark component, illustrating rich soliton structure, while the -soliton construction provides a scalable framework controlled by spectral data and mixing matrices. The results offer strong evidence for integrability of the constrained model and show practical applicability to gradient NLSEs, motivating further study of conservation laws and full inverse scattering formulations.

Abstract

We consider the linear vector Schrödinger equation subjected to quadratic constraints. We demonstrate that the resulting nonlinear system is closely related to the Ablowitz-Ladik hierarchy and use this fact to derive the N-soliton solutions for the discussed model. As an example of application of these results we present solitons of some vector nonlinear Schrödinger equation with gradient nonlinearity.

Paper Structure

This paper contains 5 sections, 3 theorems, 102 equations.

Table of Contents

  1. Introduction.
  2. Ablowitz-Ladik hierarchy.
  3. $N$-soliton solutions.
  4. Vector NLSE with gradient nonlinearity.
  5. Conclusion.

Key Result

Proposition 3.1

Vectors $\boldsymbol{q}$ and $\boldsymbol{r}$ defined in (def:qr) and (sol:q1)--(sol:q4) with where $\langle a_{0} | = \left( a_{01}, ... , a_{0N} \right)$ and $\langle b_{0} | = \left( b_{01}, ... , b_{0N} \right)$ are arbitrary constant $N$-rows, and satisfy equations (nlse:q)--(eq:lambda) and constraint (uni:qr).

Theorems & Definitions (3)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1