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Potential well theory for the derivative nonlinear Schrödinger equation

Masayuki Hayashi

TL;DR

This work analyzes the derivative nonlinear Schrödinger equation $i\partial_t u + \partial_x^2 u + i|u|^2\partial_x u + b|u|^4 u=0$, revealing a mass-critical structure and a Hamiltonian framework that persists for general $b$. By employing gauge transformations and a two-parameter variational/potential-well approach, the authors establish a mass threshold $M^*(b)$ that governs global well-posedness and delineates a turning point in the soliton-induced potential wells; they construct a two-parameter family of potential wells whose invariance and bifurcation properties encode stability and global dynamics. They develop a comprehensive soliton theory, provide explicit soliton profiles, mass, and momentum formulas, and connect the threshold to both DNLS and velocity-parameter solitons, including algebraic solitons at critical velocities. In the Hamiltonian (DNLS) form, they derive a sharp $4\pi$-mass condition and relate algebraic solitons to the boundary of potential wells, yielding a unified variational picture beyond complete integrability. The results advance understanding of long-time dynamics, global existence, and soliton-driven thresholds for derivative NLS models in one dimension.

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 known as a gauge equivalent form of well-known derivative nonlinear Schrödinger equation (DNLS), which is mass critical and completely integrable. The equation can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data $u_0\in H^1(\mathbb{R})$ satisfies the mass condition $\| u_0\|_{L^2}^2 <4π$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation for general $b\in\mathbb{R}$, which is exactly corresponding to $4π$-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both $4π$-mass condition and algebraic solitons.

Potential well theory for the derivative nonlinear Schrödinger equation

TL;DR

This work analyzes the derivative nonlinear Schrödinger equation , revealing a mass-critical structure and a Hamiltonian framework that persists for general . By employing gauge transformations and a two-parameter variational/potential-well approach, the authors establish a mass threshold that governs global well-posedness and delineates a turning point in the soliton-induced potential wells; they construct a two-parameter family of potential wells whose invariance and bifurcation properties encode stability and global dynamics. They develop a comprehensive soliton theory, provide explicit soliton profiles, mass, and momentum formulas, and connect the threshold to both DNLS and velocity-parameter solitons, including algebraic solitons at critical velocities. In the Hamiltonian (DNLS) form, they derive a sharp -mass condition and relate algebraic solitons to the boundary of potential wells, yielding a unified variational picture beyond complete integrability. The results advance understanding of long-time dynamics, global existence, and soliton-driven thresholds for derivative NLS models in one dimension.

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 known as a gauge equivalent form of well-known derivative nonlinear Schrödinger equation (DNLS), which is mass critical and completely integrable. The equation can be considered as a generalized equation of DNLS while preserving mass criticality and Hamiltonian structure. For DNLS it is known that if the initial data satisfies the mass condition , the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation for general , which is exactly corresponding to -mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both -mass condition and algebraic solitons.

Paper Structure

This paper contains 16 sections, 28 theorems, 203 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. DNLS and mass critical NLS
  4. Potential well theory for NLS
  5. Two types of solitons
  6. Main results
  7. Organization of the paper
  8. Solitons and conserved quantities
  9. A two-parameter family of solitons
  10. Mass of the solitons
  11. Momentum of the solitons
  12. Positivity of the momentum
  13. Gauge transformation and local well-posedness in $H^1({\mathbb R})$
  14. Variational characterization
  15. A two-parameter family of potential wells
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $b<0$. The equation eq:1.1 has a two-parameter family of solitons $u_{\omega ,c}(x,t)=e^{i\omega t}\phi_{\omega ,c}(x-ct)$ if and only if $(\omega ,c)$ satisfies where $s_{\ast} :=\sqrt{ -\gamma /(1-\gamma)}$ and $\gamma :=1+\frac{16}{3}b$.

Figures (3)

  • Figure 1: The stable/unstable region of solitons in the case $b>0$.
  • Figure 2: Existence region of solitons.
  • Figure 3: The function $s\mapsto P(\phi_{1,2s})$ for several values of $b>-3/16$.

Theorems & Definitions (51)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 41 more