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Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach

Masayuki Hayashi

TL;DR

This work analyzes a derivative-type nonlinear Schrödinger equation with a quintic perturbation and proves orbital stability for its two-parameter soliton family when $-\tfrac{3}{16}<b<0$, including algebraic solitons that arise at $c=2\sqrt{\omega}$. The authors establish a close link between algebraic and exponential solitons (Theorem 1.1) by showing algebraic solitons are strong limits of exponential ones along a scaling curve, and prove stability via a variational framework on Nehari manifolds and potential wells (Theorem 1.2), complemented by a Cazenave–Lions type argument for the $b\le -\tfrac{3}{16}$ regime with negative velocity (Theorem 1.4). The analysis leverages the gauge-equivalent form, a two-parameter soliton family, and careful use of scaling, mass constraints, and concentration-compactness-type tools to handle both algebraic and exponentially decaying solitons within a unified variational approach. These results provide, for the first time in this context, a rigorous example of stable algebraic solitons in a two-parameter derivative NLS model and offer insights into how soliton stability interacts with the underlying variational structure and momentum behavior.

Abstract

We consider the following nonlinear Schrödinger equation of derivative type: \begin{equation} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in \mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of the DNLS equation satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in the equation only affects the coefficient in front of the quintic term in the elliptic equation, so the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If $b>-\frac{3}{16}$, the equation has algebraic solitons as well as exponentially decaying solitons. In this paper we study stability properties of solitons by variational approach, and prove that if $b<0$, all solitons including algebraic solitons are stable in the energy space. The existence of stable algebraic solitons shows an interesting mathematical example because stable algebraic solitons are not known in the context of the corresponding double power NLS.

Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach

TL;DR

This work analyzes a derivative-type nonlinear Schrödinger equation with a quintic perturbation and proves orbital stability for its two-parameter soliton family when , including algebraic solitons that arise at . The authors establish a close link between algebraic and exponential solitons (Theorem 1.1) by showing algebraic solitons are strong limits of exponential ones along a scaling curve, and prove stability via a variational framework on Nehari manifolds and potential wells (Theorem 1.2), complemented by a Cazenave–Lions type argument for the regime with negative velocity (Theorem 1.4). The analysis leverages the gauge-equivalent form, a two-parameter soliton family, and careful use of scaling, mass constraints, and concentration-compactness-type tools to handle both algebraic and exponentially decaying solitons within a unified variational approach. These results provide, for the first time in this context, a rigorous example of stable algebraic solitons in a two-parameter derivative NLS model and offer insights into how soliton stability interacts with the underlying variational structure and momentum behavior.

Abstract

We consider the following nonlinear Schrödinger equation of derivative type: \begin{equation} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in \mathbb{R}. \end{equation} If , this equation is a gauge equivalent form of well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of the DNLS equation satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in the equation only affects the coefficient in front of the quintic term in the elliptic equation, so the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If , the equation has algebraic solitons as well as exponentially decaying solitons. In this paper we study stability properties of solitons by variational approach, and prove that if , all solitons including algebraic solitons are stable in the energy space. The existence of stable algebraic solitons shows an interesting mathematical example because stable algebraic solitons are not known in the context of the corresponding double power NLS.

Paper Structure

This paper contains 13 sections, 22 theorems, 161 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Solitons
  4. Statement of the results
  5. Comments on the main result
  6. Stability results for the case $b\le -\frac{3}{16}$
  7. Organization of the paper
  8. Preliminaries
  9. Connection between two types of the solitons
  10. Stability of two types of solitons
  11. Variational characterization
  12. Stability theory with potential wells
  13. Stability of solitons with negative velocity

Key Result

Theorem 1.1

Let $b>-\frac{3}{16}$. Suppose that $(\omega_0 ,c_0)$ satisfies $c_0=2\sqrt{\omega_0}$. Then, we have for any $m\in{\mathbb Z}_{\geq 0}$.

Figures (1)

  • Figure 1: The stable/unstable region of solitons in the case $b>0$.

Theorems & Definitions (39)

  • Definition
  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Proposition 1.3: CO06O14
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • ...and 29 more