The Linear KdV Equation with an Interface

Abstract

The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.

The Korteweg–de Vries equation on a metric star graph

“…In this sense, Holmer [21] describes boundary conditions for IBVPs associated to the KdV equation on the positive and negative half-lines that imply a result of well-posedness. For other nice discussion about the boundary condition for KdV on half-lines, based on the behavior of characteristic curves we refer the reader to [17].…”

The linearised Korteweg–deVries equation on general metric graphs

“…If the coefficients (α e ) and (β e ) are constant, we can interpret the equation as the linearized KdV equation on the real line with a generalized point interaction at 0 (which corresponds to the vertex v). This situation was considered in [2].…”

“…However, the focus was put on Schrödinger type operators and the corresponding heat and Schrödinger evolution equations, see [1] and references therein. Recently, also KdV-type equations on star graphs have gained interest, see [2,9,7,8,5]. Such star graphs give rise to model either singular interactions at one point, i.e.…”

The time-dependent Schrödinger equation with piecewise constant potentials

“…Recently, the UTM has been used to construct explicit closed-form solutions of classical interface problems [4,7,8,30,31,33]. These are initial-boundary value problems for partial differential equations (PDEs) for which the solution of an equation in one domain prescribes boundary conditions for the equation in adjacent domains.…”