Separation of the Initial Conditions in the Inverse Problem for One‐Dimensional Nonlinear Tsunami Wave Run‐Up Theory.

We investigate the inverse tsunami wave problem within the framework of the one‐dimensional (1D) nonlinear shallow water equations (SWE). Specifically, we show that the initial displacement η0(x)$\eta _0(x)$ and velocity u0(x)$u_0(x)$ of the wave can be recovered, given the known motion of the shoreline R(t)$R(t)$ (the wet/dry free boundary), in terms of the Abel transform. We demonstrate that for power‐shaped inclined bathymetries, this problem admits a complete solution for any η0$\eta _0$ and u0$u_0$, provided the wave does not break. It is important to note that, in contrast to the direct problem (also known as the tsunami wave run‐up problem), where R(t)$R(t)$ can be computed exactly only for u0(x)=0$u_0(x)=0$, our algorithm can recover η0$\eta _0$ and u0$u_0$ exactly for any non‐zero u0$u_0$. This highlights an interesting asymmetry between the direct and inverse problems. Our results extend the work presented earlier, where the inverse problem was solved for u0(x)=0$u_0(x)=0$. 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. [ABSTRACT FROM AUTHOR]