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Blow-up criteria for the 1D NLS with combined nonlinearities

Alex D Rodriguez

Abstract

In this paper we develop two different types of criteria for the finite time blow-up solutions to the combined nonlinear Schrödinger equation in 1D. The first one is a negative energy criterion developed for triple combined nonlinearity and then for a (convergent) infinite sum of combined nonlinearities, including an exponential nonlinearity. The second one is a positive energy criterion for the double nonlinearity with the defocusing-focusing coefficients. We also provide examples of initial data that satisfy either of the criteria.

Blow-up criteria for the 1D NLS with combined nonlinearities

Abstract

In this paper we develop two different types of criteria for the finite time blow-up solutions to the combined nonlinear Schrödinger equation in 1D. The first one is a negative energy criterion developed for triple combined nonlinearity and then for a (convergent) infinite sum of combined nonlinearities, including an exponential nonlinearity. The second one is a positive energy criterion for the double nonlinearity with the defocusing-focusing coefficients. We also provide examples of initial data that satisfy either of the criteria.
Paper Structure (23 sections, 5 theorems, 185 equations, 5 figures, 1 table)

This paper contains 23 sections, 5 theorems, 185 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mass, energy, and the virial argument
  4. Initial data with quadratic phase
  5. Blow-up for three nonlinearities
  6. Proof of Theorem \ref{['NegE_Bup_3term']}
  7. Blow-up for an infinite sum
  8. Proof of Theorem \ref{['NegE_Bup_Exp']}
  9. A variation of the technique.
  10. A proof of Corollary \ref{['Generalization']}
  11. The special case.
  12. Blow-up with positive energy
  13. Useful Inequalities
  14. Proof of Theorem \ref{['DR_Bup']}
  15. Critical $p=5$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $u_0 \in H^1(\mathbb{R})$ such that $V[u_0] < \infty$. Let $u(x,t)$ be a solution to ClassicNLS with the given initial condition $u_0$ and the nonlinearity such that $\lambda_3 < 0$, $\alpha_3>4$ (the largest nonlinearity is always focusing and greater than 4), $\lambda_1, \lambda_2 \in \mathbb R$ and $\alpha_1,\alpha_2 \in \mathbb R^+$. Assume for each constant $C(\delta)>0$ (different in ea

Figures (5)

  • Figure 1: Sketch of $U(B)$ with maximum at $B_{\text{max}}$
  • Figure 2: The inequality \ref{['SubCondition2_exp']} holds inside the blue area, with $\theta \in (0,10)$ and $A \in (0,10)$.
  • Figure 3: Visualization of criteria \ref{['Weighted_specific_example']}, where $\theta \in (0,50)$ and $A \in (0,10)$. The inequality \ref{['Weighted_specific_example']} holds inside the blue area.
  • Figure 4: The blue area shows the range, for which $A$ ensures positive energy (i.e., \ref{['E:PosE_p5q7']}$> 0$), while the orange area (which covers the blue and extends further out) shows the range of $A$ such that \ref{['p5q7_criteria']} holds. Notice the orange region has overlap.
  • Figure 5: The blue area shows the range for which $A$ ensures positive energy (i.e., \ref{['E_poly']}$>0$), while the orange area (on top of blue) shows the range of $A$ such that \ref{['Poly_bup']} holds.

Theorems & Definitions (15)

  • Theorem 1.1: Negative energy blow-up for three nonlinear terms
  • Theorem 1.2: Negative energy blow-up for exponential nonlinearity
  • Remark 1.3
  • Corollary 1.4: Negative energy blow-up for finite or infinite number of nonlinear terms
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Positive energy blow up for two nonlinear terms
  • Remark 1.9
  • Remark 1.10
  • ...and 5 more