# Solvability of the initial value problem to the Isobe-Kakinuma model for water waves

## TL;DR

This paper proves local-in-time solvability of the initial value problem for the Isobe-Kakinuma water wave model, a nonlinear dispersive PDE system, under specific initial data restrictions and sign conditions.

## Contribution

It establishes the existence of solutions for the Isobe-Kakinuma model with initial data restrictions and analyzes the model's linear dispersion relation.

## Key findings

- Initial value problem is solvable locally in time under certain conditions.
- The initial data must satisfy an infinite-dimensional manifold restriction.
- The model's linear dispersion relation is characterized.

## Abstract

We consider the initial value problem to the Isobe-Kakinuma model for water waves and the structure of the model. The Isobe-Kakinuma model is the Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The Isobe-Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations. Since the hypersurface $t=0$ is characteristic for the Isobe-Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh-Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces. We also discuss the linear dispersion relation to the model.