The KdV equation on the half-line: The Dirichlet to Neumann map

TL;DR

This paper analyzes the initial-boundary value problem for the KdV equation on the half-line, providing a systematic way to determine unknown boundary data via nonlinear integral equations, facilitating solutions in various boundary conditions.

Contribution

It introduces a method to characterize unknown boundary values for the KdV equation using nonlinear integral equations, applicable to Dirichlet and Neumann problems.

Findings

Effective integral equations can be solved perturbatively to all orders.

Characterizations apply to both Dirichlet and Neumann boundary conditions.

The approach provides a recursive scheme for boundary value determination.

Abstract

We consider initial-boundary value problems for the KdV equation $u_t + u_x + 6uu_x + u_{xxx} = 0$ on the half-line $x \geq 0$. For a well-posed problem, the initial data $u(x,0)$ as well as one of the three boundary values $\{u(0,t), u_x(0,t), u_{xx}(0,t)\}$ can be prescribed; the other two boundary values remain unknown. We provide a characterization of the unknown boundary values for the Dirichlet as well as the two Neumann problems in terms of a system of nonlinear integral equations. The characterizations are effective in the sense that the integral equations can be solved perturbatively to all orders in a well-defined recursive scheme.