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Strong stabilization of damped nonlinear Schr{ö}dinger equation with saturation on unbounded domains

Pascal Bégout, Jesús Ildefonso Díaz

Abstract

We consider the damped nonlinear Schr\''{o}dinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term $V(x)u,$ a nonlinear term of the form $\mathrm{i}μu(t,x)/|u(t,x)|$ for a given parameter $μ>0$ (arising in optical applications on non-Kerr-like fibers). In the right hand side we assume a given forcing term $f(t,x).$ The important new difficulty, in contrast to previous results in the literature, comes from the fact that the spatial domain is assumed to be unbounded. We start by proving the existence and uniqueness of weak and strong solutions according the regularity of the data of the problem. The existence of solutions with a lower regularity is also obtained by working with a sequence of spaces verifying the Radon-Nikodým property. Concerning the asymptotic behavior for large times we prove a strong stabilization result. For instance, in the one dimensional case we prove that there is extinction in finite time of the solutions under the mere assumption that the $L^\infty$-norm of the forcing term $f(t,x)$ becomes less than $μ$ after a finite time. This presents some analogies with the so called feedback \textit{bang-bang controls} $v$ (here $v=-\mathrm{i}μu/|u|+f).$

Strong stabilization of damped nonlinear Schr{ö}dinger equation with saturation on unbounded domains

Abstract

We consider the damped nonlinear Schr\''{o}dinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term a nonlinear term of the form for a given parameter (arising in optical applications on non-Kerr-like fibers). In the right hand side we assume a given forcing term The important new difficulty, in contrast to previous results in the literature, comes from the fact that the spatial domain is assumed to be unbounded. We start by proving the existence and uniqueness of weak and strong solutions according the regularity of the data of the problem. The existence of solutions with a lower regularity is also obtained by working with a sequence of spaces verifying the Radon-Nikodým property. Concerning the asymptotic behavior for large times we prove a strong stabilization result. For instance, in the one dimensional case we prove that there is extinction in finite time of the solutions under the mere assumption that the -norm of the forcing term becomes less than after a finite time. This presents some analogies with the so called feedback \textit{bang-bang controls} (here
Paper Structure (8 sections, 18 theorems, 142 equations)

This paper contains 8 sections, 18 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Existence and uniqueness of weak and strong solutions
  3. Finite time extinction and asymptotic behavior for large time
  4. On the non-measurability
  5. Some results of Functional Analysis
  6. Proofs of the theorems on uniqueness and existence of strong and weak solutions
  7. Proofs of the finite time extinction and the asymptotic behavior theorems
  8. Solutions with lower regularity

Key Result

Proposition 2.4

Assume O, V and pV. Let $\mu\in\mathbb{C},$$f\in L^1_\mathrm{loc}([0,\infty);L^2(\Omega))$ and $u_0\in L^2(\Omega).$ Let $(Y_n)_{n\in\mathbb{N}_0}$ be any $L^1$-approximating sequence of RNP-spaces $($see Definition defRNP below$).$ If $u$ is a weak solution to nls--u0 then for any $n\in\mathbb{N}_0 and $u$ solves nls in $L^1_\mathrm{loc}([0,\infty);H^{-2}(\Omega)+Y^\star_n)\hookrightarrow\mathscr

Theorems & Definitions (29)

  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6: Uniqueness and continuous dependence
  • Theorem 2.7: Existence and uniqueness of weak solutions
  • Remark 2.8
  • Theorem 2.9: Existence and uniqueness of strong solutions
  • Remark 2.10
  • Theorem 3.1
  • ...and 19 more