The Small-Dispersion Limit of the Intermediate Long Wave Equation via Semiclassical Soliton Ensembles
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914313541779456 |
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| author | Mitchell, Matthew Dominique |
| author_facet | Mitchell, Matthew Dominique |
| contents | We study the small dispersion limit of the intermediate long wave (ILW) equation, specifically on a class of well-behaved initial conditions $u_0$ where the number of solitons in the solution increases without bound. First, we conduct a formal WKB-style analysis on the ILW direct scattering problem, generating approximate eigenvalues and norming constants. We then use this to define a modified set of scattering data and rigorously analyze the associated inverse scattering problem. The main results include demonstrating $L^2$-convergence of the solution at $t = 0$ to the original initial condition $u_0$ and for $0 < t < t_\mathrm{c}$ to the associated solution of invicid Burgers' equation, where $t_\mathrm{c}$ is the time of gradient catastrophe. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Small-Dispersion Limit of the Intermediate Long Wave Equation via Semiclassical Soliton Ensembles Mitchell, Matthew Dominique Analysis of PDEs Mathematical Physics 35Q51 31A10 30E25 We study the small dispersion limit of the intermediate long wave (ILW) equation, specifically on a class of well-behaved initial conditions $u_0$ where the number of solitons in the solution increases without bound. First, we conduct a formal WKB-style analysis on the ILW direct scattering problem, generating approximate eigenvalues and norming constants. We then use this to define a modified set of scattering data and rigorously analyze the associated inverse scattering problem. The main results include demonstrating $L^2$-convergence of the solution at $t = 0$ to the original initial condition $u_0$ and for $0 < t < t_\mathrm{c}$ to the associated solution of invicid Burgers' equation, where $t_\mathrm{c}$ is the time of gradient catastrophe. |
| title | The Small-Dispersion Limit of the Intermediate Long Wave Equation via Semiclassical Soliton Ensembles |
| topic | Analysis of PDEs Mathematical Physics 35Q51 31A10 30E25 |
| url | https://arxiv.org/abs/2602.03063 |