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Resonant Solutions of the Non-linear Schroedinger Equation with Periodic Potential

A. Duaibes, Yu. Karpeshina

TL;DR

The work develops a high-energy perturbation framework for the nonlinear Schrödinger equation with a periodic potential in two dimensions, proving the existence of stationary states near two-plane-wave superpositions. It blends Bloch theory for the linear polyharmonic operator with a nonlinear fixed-point scheme to incorporate the nonlinearity, yielding convergent series for perturbed eigenvalues and spectral projections and explicit control of isoenergetic surfaces. In resonant regimes, a 2×2 model captures near-degeneracies, and the authors construct nonlinear corrections producing paired eigenfunctions $u^{\pm}$ with precise high-energy asymptotics and differentiability with respect to the Bloch parameter. Specializing to $n=2$, $l=1$, the results guarantee resonant nonlinear Bloch states for sufficiently small $V$ and small amplitudes, with explicit error bounds and separation of energy branches, enabling detailed geometry of isoenergetic surfaces at large energies.

Abstract

The goal is a construction of stationary solutions close to a non-trivial combination of two plane waves at high energies for a periodic non-linear Schroedinger equation in dimension two. The corresponding isoenergetic surfaces are described for every sufficiently large energy.

Resonant Solutions of the Non-linear Schroedinger Equation with Periodic Potential

TL;DR

The work develops a high-energy perturbation framework for the nonlinear Schrödinger equation with a periodic potential in two dimensions, proving the existence of stationary states near two-plane-wave superpositions. It blends Bloch theory for the linear polyharmonic operator with a nonlinear fixed-point scheme to incorporate the nonlinearity, yielding convergent series for perturbed eigenvalues and spectral projections and explicit control of isoenergetic surfaces. In resonant regimes, a 2×2 model captures near-degeneracies, and the authors construct nonlinear corrections producing paired eigenfunctions with precise high-energy asymptotics and differentiability with respect to the Bloch parameter. Specializing to , , the results guarantee resonant nonlinear Bloch states for sufficiently small and small amplitudes, with explicit error bounds and separation of energy branches, enabling detailed geometry of isoenergetic surfaces at large energies.

Abstract

The goal is a construction of stationary solutions close to a non-trivial combination of two plane waves at high energies for a periodic non-linear Schroedinger equation in dimension two. The corresponding isoenergetic surfaces are described for every sufficiently large energy.
Paper Structure (9 sections, 43 theorems, 197 equations)

This paper contains 9 sections, 43 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Linear Polyharmonic Equation with Periodic Potential
  4. Nonlinear Periodic Polyharmonic Equation
  5. Perturbation Theory for Non-linear Polyharmonic Equation with Periodic Potential for $2l>n$. Resonant Case.
  6. Construction of a Cauchy Sequence
  7. Solution to Nonlinear Polyharmonic Equation with Periodic Potential for $2l>n$
  8. The Differentiability of the Eigenvalue and Its Spectral Projection
  9. Solutions of Nonlinear Scrödinger Equation in Dimension Two.

Key Result

Lemma 1

For an arbitrarily small positive, $\delta$, $2\delta<2l-n$, and sufficiently large $k>k_0(\delta )$, there exists a non-resonant set $\chi _0(k,\delta )$, belonging to the isoenergetic surface $S_{0}(k)$ of the free operator $H_{0}(t)$, such that, for any $\vec{t}$ in it, Moreover, the nonsingular set has an asymptotically full measure on $S_{0}(k)$: where $s(.)$ denotes the Lebesgue measure.

Theorems & Definitions (60)

  • Lemma 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Definition 1
  • Lemma 7
  • Corollary 8
  • Theorem 9
  • ...and 50 more