[Paper Review] Space Quasi-Periodic Standing Waves for Nonlinear Schrödinger Equations

W.-M. Wang|arXiv (Cornell University)|Jun 6, 2018

Nonlinear Photonic Systems16 references3 citations

TL;DR

This paper establishes the existence of space quasi-periodic standing wave solutions for the nonlinear Schrödinger equation in $¹\mathbb{R}^d$ via a KAM-type iterative scheme. For small-amplitude solutions bifurcating from linear generalized eigenfunctions, it proves that such solutions exist for a positive-measure set of frequency parameters $µ\lambda$, with the nonlinear frequency corrected by $\mathcal{O}(|a|^{2p})$ and the solution itself perturbed by $\mathcal{O}(|a|^p)$, valid in arbitrary dimensions.

ABSTRACT

We construct space quasi-periodic standing wave solutions to the nonlinear Schrödinger equations on R^d for arbitrary d. This is a type of quasi-periodic nonlinear Bloch-Floquet waves.

Motivation & Objective

• To construct global, non-decaying, space quasi-periodic solutions to the nonlinear Schrödinger equation (NLS) in $\mathbb{R}^d$ with no underlying translation symmetry.
• To extend the existence of such solutions beyond periodic or localized settings, particularly in the non-compact $\mathbb{R}^d$ setting.
• To establish that every small even generalized eigenfunction of the linear Schrödinger operator bifurcates into a solution of the nonlinear equation under appropriate conditions on the frequency parameters $\lambda_k$.
• To provide a rigorous measure-theoretic construction of the set of admissible $\lambda$ parameters via a KAM-type iterative scheme, yielding solutions that are quasi-periodic in space and periodic in time.

Proposed method

• Uses a KAM-type iterative scheme to solve the stationary nonlinear elliptic equation $-\Delta u - |u|^{2p}u = Eu$ derived from the NLS standing wave ansatz $U(t,x) = e^{-iEt}u(x)$.
• Employs a Newton-type iteration to construct solutions as quasi-periodic cosine series in $x_k$, with frequencies $\lambda_k \in (1/2, 3/2)^2$ and $j_k \in \mathbb{Z}^2$, ensuring non-resonance via Diophantine-type conditions.
• Applies a sequence of nested Cantor-like sets $\Lambda_r$ in the parameter space, with measure estimates $\text{meas }\Lambda \geq 1 - |a|^{p/6}$ for small $|a|$, ensuring convergence of the iteration.
• Implements resolvent estimates and inverse bounds on finite-volume operators $T_M$, using exponential decay estimates on the inverse kernel to control the nonlinear terms.
• Uses pointwise decay estimates on the inverse of the linearized operator, with parameters $\alpha$ and $\delta$ adjusted at each step to maintain control under perturbations.
• Applies a bootstrapping argument to verify that the solution sequence satisfies the required decay, regularity, and derivative bounds at each stage of the iteration.

Experimental results

Research questions

• RQ1 Can space quasi-periodic standing wave solutions exist for the nonlinear Schrödinger equation in $\mathbb{R}^d$ without decay or periodicity in space?
• RQ2 Does every small-amplitude even generalized eigenfunction of the linear Schrödinger operator bifurcate into a solution of the nonlinear equation for a positive-measure set of frequency parameters $\lambda$?
• RQ3 Can a KAM-type iterative method be adapted to construct such solutions in the non-compact $\mathbb{R}^d$ setting, where standard compactness or periodicity assumptions fail?
• RQ4 What is the measure of the set of frequency parameters $\lambda$ for which such solutions exist, and how does it depend on the amplitude $a$?
• RQ5 How does the nonlinear frequency $E$ depend on $\lambda$, and is it smooth?

Key findings

• For every small-amplitude linear solution of the form $\tilde{U} = a e^{-i(\tilde{j} \cdot \lambda)^2 t} \prod_{k=1}^d \cos(\tilde{j}_k \cdot \lambda_k x_k)$, there exists a nonlinear solution $U$ bifurcating from it, with the same spatial quasi-periodic structure.
• The solution $U(t,x)$ is of the form $e^{-i[(\tilde{j} \cdot \lambda)^2 + \mathcal{O}(|a|^{2p})]t} \left[ a \prod_{k=1}^d \cos(\tilde{j}_k \cdot \lambda_k x_k) + \mathcal{O}(|a|^p) \right]$, showing a small correction to the amplitude and frequency.
• The set of admissible $\lambda$ parameters, denoted $\Lambda$, has measure satisfying $\text{meas }\Lambda \geq 1 - |a|^{p/6}$, which is positive for small $|a|$, and is a Cantor-like set of positive measure.
• The nonlinear eigenvalue $E$ is $C^1$ in $\lambda$ on $(1/2, 3/2)^{2d}$, indicating smooth dependence on the frequency parameters.
• The construction is valid in arbitrary dimensions $d$, and the method extends to higher-dimensional $\lambda_k$ with the same proof structure.
• The solution is time-periodic (with one basic frequency), space quasi-periodic, and globally defined in time, with no localization or underlying translation symmetry.