Separation of the initial conditions in the inverse problem for 1D non-linear tsunami wave run-up theory
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arXiv
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| Format: | Preprint |
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2025
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| author | Rybkin, Alexei Bobrovnikov, Oleksandr Palmer, Noah Abramowicz, Daniel Pelinovsky, Efim |
| author_facet | Rybkin, Alexei Bobrovnikov, Oleksandr Palmer, Noah Abramowicz, Daniel Pelinovsky, Efim |
| contents | We investigate the inverse tsunami wave problem within the framework of the 1D nonlinear shallow water equations (SWE). Specifically, we focus on determining the initial displacement $η_0(x)$ and velocity $u_0(x)$ of the wave, given the known motion of the shoreline $R(t)$ (the wet/dry free boundary). We demonstrate that for power-shaped inclined bathymetries, this problem admits a complete solution for any $η_0$ and $u_0$, provided the wave does not break. In particular, we show that the knowledge of $R(t)$ enables the unique recovery of both $η_0(x$) and $u_0(x)$ in terms of the Abel transform.
It is important to note that, in contrast to the direct problem (also known as the tsunami wave run-up problem), where $R(t)$ can be computed exactly only for $u_0(x)=0$, our algorithm can recover $η_0$ and $u_0$ exactly for any non-zero $u_0$. This highlights an interesting asymmetry between the direct and inverse problems. Our results extend the work presented in \cite{Rybkin23,Rybkin24}, where the inverse problem was solved for $u_0(x)=0$. As in previous work, our approach utilizes the Carrier-Greenspan transformation, which linearizes the SWE for inclined bathymetries. Extensive numerical experiments confirm the efficiency of our algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Separation of the initial conditions in the inverse problem for 1D non-linear tsunami wave run-up theory Rybkin, Alexei Bobrovnikov, Oleksandr Palmer, Noah Abramowicz, Daniel Pelinovsky, Efim Analysis of PDEs We investigate the inverse tsunami wave problem within the framework of the 1D nonlinear shallow water equations (SWE). Specifically, we focus on determining the initial displacement $η_0(x)$ and velocity $u_0(x)$ of the wave, given the known motion of the shoreline $R(t)$ (the wet/dry free boundary). We demonstrate that for power-shaped inclined bathymetries, this problem admits a complete solution for any $η_0$ and $u_0$, provided the wave does not break. In particular, we show that the knowledge of $R(t)$ enables the unique recovery of both $η_0(x$) and $u_0(x)$ in terms of the Abel transform. It is important to note that, in contrast to the direct problem (also known as the tsunami wave run-up problem), where $R(t)$ can be computed exactly only for $u_0(x)=0$, our algorithm can recover $η_0$ and $u_0$ exactly for any non-zero $u_0$. This highlights an interesting asymmetry between the direct and inverse problems. Our results extend the work presented in \cite{Rybkin23,Rybkin24}, where the inverse problem was solved for $u_0(x)=0$. As in previous work, our approach utilizes the Carrier-Greenspan transformation, which linearizes the SWE for inclined bathymetries. Extensive numerical experiments confirm the efficiency of our algorithms. |
| title | Separation of the initial conditions in the inverse problem for 1D non-linear tsunami wave run-up theory |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.07793 |