Detecting eigenvalues in a fourth-order nonlinear Schrödinger equation with a non-regular Maslov box
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| Format: | Preprint |
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2024
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| _version_ | 1866916770826158080 |
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| author | Curran, Mitchell Marangell, Robert |
| author_facet | Curran, Mitchell Marangell, Robert |
| contents | We use the Maslov index to study the eigenvalue problem arising from the linearisation about solitons in the fourth-order cubic nonlinear Schrödinger equation (NLSE). Our analysis is motivated by recent work by Bandara et al., in which the fourth-order cubic NLSE was shown to support infinite families of multipulse solitons. Using a homotopy argument, we prove that the Morse indices of two selfadjoint fourth-order operators appearing in the linearisation may be computed by counting conjugate points, as well as a lower bound for the number of real unstable eigenvalues of the linearisation. We also give a Vakhitov-Kolokolov type stability criterion. The interesting aspects of this problem as an application of the Maslov index are the instances of non-regular crossings, which feature crossing forms with varying ranks of degeneracy. We handle such degeneracies directly via higher order crossing forms, using a definition of the Maslov index developed by Piccione and Tausk. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16903 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Detecting eigenvalues in a fourth-order nonlinear Schrödinger equation with a non-regular Maslov box Curran, Mitchell Marangell, Robert Spectral Theory 34L05, 53D12, 35B35, 37K45, 37K40, 47A75 We use the Maslov index to study the eigenvalue problem arising from the linearisation about solitons in the fourth-order cubic nonlinear Schrödinger equation (NLSE). Our analysis is motivated by recent work by Bandara et al., in which the fourth-order cubic NLSE was shown to support infinite families of multipulse solitons. Using a homotopy argument, we prove that the Morse indices of two selfadjoint fourth-order operators appearing in the linearisation may be computed by counting conjugate points, as well as a lower bound for the number of real unstable eigenvalues of the linearisation. We also give a Vakhitov-Kolokolov type stability criterion. The interesting aspects of this problem as an application of the Maslov index are the instances of non-regular crossings, which feature crossing forms with varying ranks of degeneracy. We handle such degeneracies directly via higher order crossing forms, using a definition of the Maslov index developed by Piccione and Tausk. |
| title | Detecting eigenvalues in a fourth-order nonlinear Schrödinger equation with a non-regular Maslov box |
| topic | Spectral Theory 34L05, 53D12, 35B35, 37K45, 37K40, 47A75 |
| url | https://arxiv.org/abs/2411.16903 |