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

Abstract

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if $q_t$ and $q_{xxx}$ have the same sign (KdVI) or two boundary conditions if $q_t$ and $q_{xxx}$ have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if $\{q(x,0),q(0,t)\}$ and $\{q(x,0),q(0,t),q_x(0,t)\}$ are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values $\{q_x(0,t),q_{xx}(0,t)\}$ and $\{q_{xx}(0,t)\}$, respectively. We show that this can be achieved without solving for $q(x,t)$ by analysing a certain ``global relation'' which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function ${\Phi }^{(t)}(t,k)$, where ${\Phi }^{(t)}$ satisifies the $t$-part of the associated Lax pair evaluated at $x=0$. Indeed, by employing a Gelfand--Levitan--Marchenko triangular representation for ${\Phi }^{(t)}$, the global relation can be solved \emph{explicitly} for the unknown boundary values in terms of the given initial and boundary conditions and the function ${\Phi }^{(t)}$. This yields the unknown boundary values in terms of a nonlinear Volterra integral equation.

21 pages, 3 figures

References in corpus (1)

• The Nonlinear Schrödinger Equation on the Interval

Cited by in corpus (5)

• The Unified Method: I Non-Linearizable Problems on the Half-Line
• The Unified Method: II NLS on the Half-Line with $t$-Periodic Boundary Conditions
• Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval
• The Kadomtsev-Petviashvili II Equation on the Half-Plane
• An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval