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.