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

By Justin Holmer. 2005

205 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/afd28895435ac8dc2584cd2f5fb7f8982101f56b

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane (2008, background): “This result was strengthened later by Holmer [34] to include the case − 34 s 0.”
The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line (2005, methodology): “in [ 2 ]), and the techniques of this forthcoming paper may also apply here to resolve the uniqueness problem for 0 s < 12.”
Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schr\"odinger Equations (2015, background): “2)) with nonhomogeneous boundary conditions has not progressed to the same extent (see [30, 19, 20, 21, 22, 23, 45, 46, 52, 68, 69, 70, 71] and more recent work on the boundary-value problems of some other dispersive equations [55], and the references therein).”
A non-homogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain II (2003, background): “5) is also known to be well-posed in the space Hs(R+) for any s > − (3)4 [12,28].”
Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval (2018, methodology): “The Fokas method can be used to study the boundary value problems of several important integrable equations including 2 × 2 Lax pair equations, such as the KdV equation [10, 11], the nonlinear Schrödinger equation [12, 13, 14, 15] and other PDEs [2, 3, 4, 17, 21, 23, 24, 25, 29, 30, 34].”
The unified method: I. Nonlinearizable problems on the half-line (2011, methodology): “This can be achieved by employing the Deift-Zhou method [22, 23] for the long-time asymptotics [40, 41, 42, 43] and the Deift-Zhou-Venakides method [20, 21] for the zero-dispersion limit [46, 59].”
Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain (2009, background): “In [3] and [15], one can find some well-posedness results in the case where there are nonhomogeneous Dirichlet boundary conditions.”
Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation (2012, background): “According to [24, 26], ‖qj‖ H 2−j 3 (0,T ) ≤ CT ‖f ‖Y0,b .”