We prove local well-posedness of the initial-boundary value problem for the Korteweg–de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.

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The Initial-Boundary Value Problem for the Korteweg–de Vries Equation

In this paper, we implement the Fokas method in order to study initial-boundary value problems of the coupled modified Korteweg–de Vries equation formulated on the half-line, with Lax pairs involving matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution can be written in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions and , which are respectively determined by the initial values and boundary values at . Finally, the associated Dirichlet to Neumann map of the equation is analyzed in detail. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.

https://iopscience.iop.org/article/10.1088/1751-8121/aa825b/pdf

Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method

Quantum and classical integrable sine-Gordon model with defect

The possible boundary conditions consistent with the integrability of the classical sine-Gordon equation are studied. A boundary value problem on the half-line x<or=0 with local boundary condition at the origin is considered. The most general form of this boundary condition is found such that the problem be integrable. For the resulting system an infinite number of involutive integrals of motion exist. These integrals are calculated and one is identified as the Hamiltonian. The results found agree with some recent work of Ghoshal and Zamolodchikov(1994).

Integrable boundary conditions for classical sine-Gordon theory

We investigate the nonlinear stage of the modulational (or Benjamin--Feir) instability by characterizing the initial value problem for the focusing nonlinear Schrodinger (NLS) equation with nonzero boundary conditions (NZBC) at infinity. We do so using the recently formulated inverse scattering transform (IST) for this problem. While the linearization of the NLS equation ceases to be valid when the perturbations have grown sufficiently large compared to the background, the results of the IST remain valid for all times and therefore provide a convenient way to study the nonlinear stage of the modulational instability. We begin by studying the spectral problem for the Dirac operator (i.e., the first half of the Lax pair for the NLS equation) with piecewise constant initial conditions which are a generalization to NZBC of a potential well and a potential barrier. Since the scattering data uniquely determine the time evolution of the initial condition via the inverse problem, the study of these kinds of poten...

I. T. Khabibullin

Papers

Integrable initial-boundary-value problems

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