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Total positivity and spectral properties of linearized operators

John Albert, Steven Levandosky

Abstract

For a class of semilinear elliptic equations, we establish criteria that guarantee that the linearized operator associated with a solution satisfies certain spectral assumptions that are widely used in the analysis of the stability of solitary waves. The criteria only involve the symbol of the linear operator and positivity and symmetry of the solution, and can therefore be verified without an explicit formula for the solution.

Total positivity and spectral properties of linearized operators

Abstract

For a class of semilinear elliptic equations, we establish criteria that guarantee that the linearized operator associated with a solution satisfies certain spectral assumptions that are widely used in the analysis of the stability of solitary waves. The criteria only involve the symbol of the linear operator and positivity and symmetry of the solution, and can therefore be verified without an explicit formula for the solution.
Paper Structure (4 sections, 12 theorems, 56 equations, 6 figures)

This paper contains 4 sections, 12 theorems, 56 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. A class of examples
  4. Two specific examples

Key Result

Theorem 1.3

Let $\mathcal{L}$ have symbol $\alpha$ satisfying Assumption A:alpha_assumption for some $s>\frac{1}{2}$, and let $f$ satisfy Assumption A:f_assumption. Then there exists a nontrivial solution $\varphi\in H^s(\mathbb R)$ of E:traveling.

Figures (6)

  • Figure 1: Kernels $K(x)=\mathcal{F}^{-1}\left(\frac{1}{\xi^4+b\xi^2+c}\right)$ with $c=1$, $b=-1.5,0,3$.
  • Figure 2: Solutions of $\varphi""-b\varphi"+c\varphi=\varphi^3$ with $c=1$, $b=-1.5,0,3$.
  • Figure 3: In both shaded regions, solutions of \ref{['E:ODE4']} exist. The hypotheses of Theorem \ref{['T:positivity']} hold in the darker shaded region, where the kernel $K$ is strongly $PF(2)$.
  • Figure 4: The symbol $\alpha(\xi)=\xi^6+a\xi^4+b\xi^2+c$ of the operator factors into $(\xi^2+a_1^2)(\xi^2+a_2^2)(\xi^2+a_3^2)$ for real nonzero $a_i$ in the region between the two surfaces.
  • Figure 5: When $c=1$, the symbol $\alpha(\xi)=\xi^6+a\xi^4+b\xi^2+c$ of the operator $\mathcal{L}$ satisfies \ref{['E:L']} for $(a,b)$ in the two shaded regions, and factors into $(\xi^2+a_1^2)(\xi^2+a_2^2)(\xi^2+a_3^2)$ for real nonzero $a_i$ in the darker shaded region.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 3.1
  • ...and 9 more