Long time motion of NLS solitary waves in a confining potential

TL;DR

The paper analyzes the long-time motion of solitary-wave solutions for focusing generalized nonlinear Schrödinger equations in a slowly varying confining potential $V(x)$. By combining a Lyapunov–Schmidt decomposition with energy estimates, the authors control deviations from the soliton manifold and derive modulation equations for the moving soliton parameters. They prove that the soliton center $a(t)$ and momentum $p(t)$ follow Newton-type dynamics in $V(x)$ with $O(ε^2)$ corrections over a long, finite time interval (assuming initial data are $ε$-close to the soliton manifold and satisfy an initial energy bound). The analysis employs a moving frame and a symplectic decomposition $ψ = S_{a p γ}(η_μ+w)$ with $w ot J T_{η_μ} M_s$, and constructs a Lyapunov functional Λ whose upper and lower bounds yield $w = O(ε)$ and modulation parameters with $α_j = O(ε^2)$, establishing near-soliton dynamics on time scales up to $T ≤ C(ε_V√ε_h+ε^2)^{-1}$.

Abstract

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schroedinger equations with a confining, slowly varying external potential, $V(x)$. A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential $V(x)$ over a long time interval.

Figures (2)

• Figure 1.1: The trajectory $\psi(\cdot,t)$ over the soliton Manifold $\mathrm{M}_{\mathrm{s}}$.
• Figure 5.1: Orthogonal decomposition versus skew-orthogonal decomposition.

Theorems & Definitions (42)

• Remark 2.1
• Theorem 2.1
• Remark 2.2: Remark about notation
• Theorem 4.1
• proof
• Remark 4.1
• Proposition 5.1
• proof
• Corollary 5.2
• proof