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The nonlinear Schrödinger equation with combined nonlinearities in 1D

Oscar Riaño, Alex D Rodriguez, Svetlana Roudenko

Abstract

We consider the one-dimensional nonlinear Schrödinger equation $$ iu_t + u_{xx} + \mathcal{N}(u)u=0, \quad x,t \in \mathbb R, $$ with the nonlinearity term that is expressed as a sum of powers, possibly infinite: $$ \mathcal{N}(u) = \sum d_k |u|^{α_k}, \quad α_k > 0. $$ We first investigate the local well-posedness of this equation for any positive powers of $α_k$ in a certain weighted class of initial data, subset of $H^1 (\mathbb R)$. For that we use an approach of Cazenave-Naumkin [19], thus, avoiding any Strichartz estimates. Then, using the pseudo-conformal transformation, we extend the local result to the global one for the initial data with a quadratic phase. Furthermore, we investigate the asymptotic behavior of such global solutions and prove scattering for data with the quadratic phase $e^{ib|x|^2}$ with sufficiently large positive $b$, in $H^1(\mathbb R)$. One of the advantages of considering an infinite sum in the nonlinearity term is being able to consider exponential nonlinearities, such as $e^{γ|u|^{k}} u$, as well as sine or cosine nonlinearities, and obtain well-posedness in those cases, the first such result for most of those nonlinearities. To conclude, we show numerical simulations for various examples of combined nonlinearities, including the double nonlinearity and an exponential one, then investigate the behavior of solutions with positive or negative initial $b$ in a quadratic phase data. Furthermore, we also show that a ground state in the NLS equation with combined nonlinearities no longer provides a sharp threshold for global behavior such as scattering vs. finite time blow-up, instead the equation has a much richer dynamics.

The nonlinear Schrödinger equation with combined nonlinearities in 1D

Abstract

We consider the one-dimensional nonlinear Schrödinger equation with the nonlinearity term that is expressed as a sum of powers, possibly infinite: We first investigate the local well-posedness of this equation for any positive powers of in a certain weighted class of initial data, subset of . For that we use an approach of Cazenave-Naumkin [19], thus, avoiding any Strichartz estimates. Then, using the pseudo-conformal transformation, we extend the local result to the global one for the initial data with a quadratic phase. Furthermore, we investigate the asymptotic behavior of such global solutions and prove scattering for data with the quadratic phase with sufficiently large positive , in . One of the advantages of considering an infinite sum in the nonlinearity term is being able to consider exponential nonlinearities, such as , as well as sine or cosine nonlinearities, and obtain well-posedness in those cases, the first such result for most of those nonlinearities. To conclude, we show numerical simulations for various examples of combined nonlinearities, including the double nonlinearity and an exponential one, then investigate the behavior of solutions with positive or negative initial in a quadratic phase data. Furthermore, we also show that a ground state in the NLS equation with combined nonlinearities no longer provides a sharp threshold for global behavior such as scattering vs. finite time blow-up, instead the equation has a much richer dynamics.
Paper Structure (18 sections, 18 theorems, 198 equations, 14 figures, 6 tables)

This paper contains 18 sections, 18 theorems, 198 equations, 14 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Linear and nonlinear estimates
  4. The idea of exchanging $\langle x \rangle^n$ with $e^{it\partial_x^2}$
  5. Proof of local well-posedness in space ${\mathcal{X}}$
  6. Global results for the data in space $\mathcal{X}$ with quadratic phase
  7. Proof of Theorem \ref{['scatres']}
  8. Well-posedness and scattering in $H^1$
  9. Local existence results
  10. Global existence and scattering
  11. Numerical examples
  12. NLS with a low power single nonlinear term
  13. NLS with two combined nonlinearities
  14. Ground states for the NLS with double nonlinearity
  15. Time evolution of solutions to the NLS with double nonlinearity
  16. ...and 3 more sections

Key Result

Theorem 1.1

(Local well-posedness in $\mathcal{X}$) Let $\{d_k\}$ be a sequence of complex numbers and $\{\alpha_k\}$ be a sequence of positive real numbers. Consider $n,r, M$ as in deffparame. Suppose that for any $R_0>0$ the following condition holds for sequences $\{d_k\}$ and $\{\alpha_k\}$: where If $u_0 \in \mathcal{X}$ satisfies Xinf, then there exist $T>0$ and a unique solution $u \in C([-T,T], \mat

Figures (14)

  • Figure 1: Time evolution of $u_0 = \frac{A}{({1 + x^2})^{1/2}}$ under the NLS flow \ref{['NLS_single']} with parameters: $\alpha = \frac{1}{2}$, $\epsilon = 0.5$ and $A=3$.
  • Figure 1: $\mathcal{N}(u)u=-|u|^{\frac{1}{9}}u+|u|^{\frac{7}{9}}u$
  • Figure 2: Time evolution of $u_0 = \frac{A}{(1 + x^2)^{1/3}}$ under the NLS flow \ref{['NLS_single']} with parameters: $\alpha = \frac{1}{2}$, $\epsilon = 0.5$, and $A=0.1$.
  • Figure 3: Time evolution of $u_0 = \frac{A}{(1 + x^2)^{1/4}}$ under the NLS flow \ref{['NLS_single']} with parameters: $\alpha = \frac{1}{2}$, $\epsilon = 0.5$, and $A=0.1$.
  • Figure 3: $\mathcal{N}(u)u=-|u|^{\frac{1}{9}}u+|u|^{\frac{7}{9}}u$
  • ...and 9 more figures

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8: Finite number of combined nonlinearities, space $\mathcal{X}$
  • Remark 1.9
  • Corollary 1.10: Finite number of combined nonlinearities, space $H^1$
  • ...and 26 more