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On the cubic-quintic Schrödinger equation

Laura Baldelli

TL;DR

The paper addresses traveling-wave solutions with finite energy for the cubic-quintic nonlinear Schrödinger equation under non-vanishing boundary conditions in $\mathbb{R}^d$ ($d\ge2$). It reformulates the problem via scaling to a normalized nonlinearity, derives the associated energy and Lagrangian, and proves an explicit $L^\infty$ a priori bound for traveling-wave profiles with a velocity-independent constant $C_A$. A Pohozaev-type identity and standard variational tools underpin the preliminaries, while the authors develop a mountain-pass framework in approximating domains to obtain Palais-Smale sequences; however, a key boundedness step remains unresolved, preventing full-space existence results in the subsonic regime. The work thus provides rigorous bounds, clarifies the variational structure, and lays groundwork toward full existence results for cubic-quintic traveling waves, highlighting essential obstacles and guiding future analysis.

Abstract

This paper explores the cubic-quintic Schrödinger equation in the entire Euclidean space. Our objectives are twofold: first, to advance the understanding of unresolved issues related to this equation, which are well known in the extensively studied Gross-Pitaevskii equation. Second, to consolidate existing results on the cubic-quintic equation, providing partial contributions. Specifically, we determine the explicit constant for the $L^\infty$ a priori bound and establish a partial existence result for finite energy traveling waves in suitable approximate domains of $\mathbb R^d$.

On the cubic-quintic Schrödinger equation

TL;DR

The paper addresses traveling-wave solutions with finite energy for the cubic-quintic nonlinear Schrödinger equation under non-vanishing boundary conditions in (). It reformulates the problem via scaling to a normalized nonlinearity, derives the associated energy and Lagrangian, and proves an explicit a priori bound for traveling-wave profiles with a velocity-independent constant . A Pohozaev-type identity and standard variational tools underpin the preliminaries, while the authors develop a mountain-pass framework in approximating domains to obtain Palais-Smale sequences; however, a key boundedness step remains unresolved, preventing full-space existence results in the subsonic regime. The work thus provides rigorous bounds, clarifies the variational structure, and lays groundwork toward full existence results for cubic-quintic traveling waves, highlighting essential obstacles and guiding future analysis.

Abstract

This paper explores the cubic-quintic Schrödinger equation in the entire Euclidean space. Our objectives are twofold: first, to advance the understanding of unresolved issues related to this equation, which are well known in the extensively studied Gross-Pitaevskii equation. Second, to consolidate existing results on the cubic-quintic equation, providing partial contributions. Specifically, we determine the explicit constant for the a priori bound and establish a partial existence result for finite energy traveling waves in suitable approximate domains of .

Paper Structure

This paper contains 7 sections, 9 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Traveling waves in the literature
  3. A reformulation of the problem
  4. Structure of the paper
  5. Preliminaries
  6. Explicit $L^\infty$ estimates
  7. An attempt to establish existence in approximating domains

Key Result

Lemma 1

Let $\psi$ be a finite energy solution of fapro. Then the following identity holds: where and Moreover, by the definition of the Lagrangian eq:ac, we conclude that Finally, $I^c(\psi)=0$ if and only if $\psi$ is a constant function of modulus 1.

Figures (1)

  • Figure 1: Gross-Pitaevskii and Cubic-Quintic potentials

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • proof
  • Proposition 1
  • proof
  • Lemma 6
  • ...and 2 more