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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.29040 |
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| _version_ | 1866915900947431424 |
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| author | Gallo, Erika Zweck, John Latushkin, Yuri |
| author_facet | Gallo, Erika Zweck, John Latushkin, Yuri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29040 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability of Ginzburg-Landau pulses via Fredholm determinants of Birman-Schwinger operators Gallo, Erika Zweck, John Latushkin, Yuri Spectral Theory Dynamical Systems 35Q56, 65R20, 47B10 (Primary) 47G10, 37L15 (Secondary) 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. |
| title | Stability of Ginzburg-Landau pulses via Fredholm determinants of Birman-Schwinger operators |
| topic | Spectral Theory Dynamical Systems 35Q56, 65R20, 47B10 (Primary) 47G10, 37L15 (Secondary) |
| url | https://arxiv.org/abs/2603.29040 |