Soliton solutions associated with a class of third-order ordinary linear differential operators
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912352114311168 |
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| author | Aktosun, Tuncay Choque-Rivero, Abdon E. Toledo, Ivan Unlu, Mehmet |
| author_facet | Aktosun, Tuncay Choque-Rivero, Abdon E. Toledo, Ivan Unlu, Mehmet |
| contents | Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3ψ/dx^3+Q\,dψ/dx+Pψ=k^3ψ,$ where $Q$ and $P$ are the potentials in the Schwartz class and $k^3$ is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant $\mathbf N$-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Soliton solutions associated with a class of third-order ordinary linear differential operators Aktosun, Tuncay Choque-Rivero, Abdon E. Toledo, Ivan Unlu, Mehmet Exactly Solvable and Integrable Systems Mathematical Physics 34A55 34M50 35C08 Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3ψ/dx^3+Q\,dψ/dx+Pψ=k^3ψ,$ where $Q$ and $P$ are the potentials in the Schwartz class and $k^3$ is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada--Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant $\mathbf N$-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota. |
| title | Soliton solutions associated with a class of third-order ordinary linear differential operators |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics 34A55 34M50 35C08 |
| url | https://arxiv.org/abs/2412.10971 |