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Stability of Ginzburg-Landau pulses via Fredholm determinants of Birman-Schwinger operators

Erika Gallo, John Zweck, Yuri Latushkin

Abstract

We introduce a numerical method to determine the stability of stationary pulse solutions of the complex Ginzburg-Landau equation. The method involves the computation of the point spectrum of the first-order linear differential operator with matrix-valued coefficients on the real line obtained by linearizing the Ginzburg-Landau equation about a stationary pulse. Applying a general theory of Gesztesy, Latushkin, and Makarov, we show that this point spectrum is given by the set of zeros of a 2-modified Fredholm determinant of a Hilbert-Schmidt, Birman-Schwinger operator. We establish conditions which guarantee that this operator is trace class. Applying results of Bornemann on the numerical computation of Fredholm determinants, we obtain a bound on the error between the regular Fredholm determinant of the trace class operator and its numerical approximation by a matrix determinant. We verify the new numerical Fredholm determinant method for computing the point spectrum of a Ginzburg-Landau pulse by exhibiting excellent agreement with previous methods. This new approach avoids the challenge of solving the numerically stiff system of equations for the matrix-valued Jost solutions, and it opens the way for the spectral analysis of breather solutions of nonlinear wave equations, for which an Evans function does not exist.

Stability of Ginzburg-Landau pulses via Fredholm determinants of Birman-Schwinger operators

Abstract

We introduce a numerical method to determine the stability of stationary pulse solutions of the complex Ginzburg-Landau equation. The method involves the computation of the point spectrum of the first-order linear differential operator with matrix-valued coefficients on the real line obtained by linearizing the Ginzburg-Landau equation about a stationary pulse. Applying a general theory of Gesztesy, Latushkin, and Makarov, we show that this point spectrum is given by the set of zeros of a 2-modified Fredholm determinant of a Hilbert-Schmidt, Birman-Schwinger operator. We establish conditions which guarantee that this operator is trace class. Applying results of Bornemann on the numerical computation of Fredholm determinants, we obtain a bound on the error between the regular Fredholm determinant of the trace class operator and its numerical approximation by a matrix determinant. We verify the new numerical Fredholm determinant method for computing the point spectrum of a Ginzburg-Landau pulse by exhibiting excellent agreement with previous methods. This new approach avoids the challenge of solving the numerically stiff system of equations for the matrix-valued Jost solutions, and it opens the way for the spectral analysis of breather solutions of nonlinear wave equations, for which an Evans function does not exist.

Paper Structure

This paper contains 12 sections, 21 theorems, 149 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background on the numerical approximations of Fredholm determinants
  3. Linearization of the Complex Ginzburg-Landau Equation
  4. Diagonalization of the Unperturbed System
  5. The Birman-Schwinger Operator for the Perturbed Problem
  6. The Hilbert-Schmidt Kernel
  7. Lipschitz Continuity of the Kernel
  8. Main Results
  9. Numerical Results
  10. The Evans function for the hyperbolic secant pulse
  11. General Formula for the trace of $\mathcal{K}(\lambda)$
  12. Funding Statement

Key Result

Theorem 2.1

Let $\mathbf K\in L^2(\mathbb R\times \mathbb R, \mathbb C^{k\times k})$ be a Lipschitz continuous, matrix-valued kernel such that there is an $R>0$ so that for all $|x|$, $|y|>R$ for some $C, \alpha > 0$, where $\|\cdot\|$ is a matrix norm. Let $\mathcal{K} \in \mathcal{B}_2(L^2(\mathbb R,\mathbb{C}^k))$ be the Hilbert-Schmidt operator with kernel $\mathbf K$. Then $\mathcal{K} \in \mathcal{B}_1(

Figures (3)

  • Figure 1: Left: The numerically-approximated regular and $2$-modified Fredholm determinants for the sech solution of the NLSE, as compared to analytical formulae given in terms of the Evans function, evaluated along the line $\lambda(t) = t$. Right: Zoomed in version of results on the left in which the regular Fredholm determinant is computed with finer discretizations, $\Delta x$, across the pulse
  • Figure 2: Left: Behavior of the regular and $2-$modified Fredholm determinants, for the sech solution of the NLSE, compared to analytical formulae given in terms of the Evans function, as $\lambda$ approaches the edge of $\sigma_{{\rm ess}}(\mathcal{L})$ along the line $\lambda (t) = it/2$. Right: Essential spectrum (solid blue lines) and eigenvalues, for a stationary pulse solution of the CGLE, computed using the approximate regular Fredholm determinant (red pluses) compared with results obtained by Shen et al. shen2016spectra (blue circles)
  • Figure 3: Left: Approximate regular Fredholm determinant, for the stationary pulse solution of the CGLE, along the line $\lambda(t) = \lambda_ct,$ between eigenvalue $\lambda = 0$ and complex eigenvalue $\lambda_c \approx -0.0033 + 0.0704i$. Right: Behavior of the regular (dashed red line) and $2-$modified (solid blue line) Fredholm determinants as $\lambda$ approaches the edge, $\lambda_{{e}}$, of the essential spectrum along the line $\lambda(t) = \lambda_c + (\lambda_{{e}} - \lambda_c)t$

Theorems & Definitions (52)

  • Theorem 2.1: zweck2024regularity
  • Theorem 2.2: GZL2025NumericalFredholm
  • Remark 2.3
  • Theorem 2.4: GZL2025NumericalFredholm
  • Remark 2.5
  • Remark 4.1
  • Remark 4.2
  • Proposition 4.3
  • Proposition 4.5
  • proof
  • ...and 42 more