THE GENERALIZED KORTEWEG–DE VRIES EQUATION ON THE HALF LINE

By James E. Colliander, Carlos E. Kenig. 2001

173 citations recorded by the source corpus.

ABSTRACT The initial-boundary value problem for the generalized Korteweg–de Vries equation on a half-line is studied by adapting the initial value techniques developed by Kenig, Ponce and Vega and Bourgain to the initial-boundary setting. The approach consists of replacing the initial-boundary problem by a forced initial value problem. The forcing is selected to satisfy the boundary condition by inverting a Riemann–Liouville fractional integral.

Subfields: Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2d0eb38cc4cf359a85e32aed8e058600b490f697

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Citations in the corpus

Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schr\"odinger Equations (2015, methodology): “Colliander and Kenig [33] introduced a new method to analyze this problem by solving the pure 11 IVP wt + w kwx + wxxx = δ(x)f(t), x ∈ R, t ∈ (0, T ) , w(x, 0) = ψ(x), (2.15) of a forced, generalized KdV equation with an appropriate forcing function f(t).”
Non-homogeneous boundary value problems for the Korteweg–de Vries and the Korteweg–de Vries–Burgers equations in a quarter plane (2008, background): “They showed in [20] that for a given s-compatible pair (φ,h) ∈ H( +)×H s+1 3 loc ( +) with 0 s 1, s = (1)2 , the IBVP (1.”
Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation (2012, background): “The interested readers are referred to [3, 5, 7, 8, 10, 18, 20, 24, 28, 29] and the references therein for an overall review for the well-posedness of the IBVP of the KdV and KdVB equations posed a finite domain and [1, 4, 9, 15–17, 19, 24] for the IBVP of the KdV and KdVB equations posed on the hal”
An Initial Boundary-Value Problem in a Half-Strip for the Korteweg–De Vries Equation in Fractional-Order Sobolev Spaces (2005, methodology): “For a = 0 and s = 0 estimates of such a type were established in Colliander and Kenig (2002).”
On the Cauchy Problem for the Generalized Korteweg-de Vries Equation (2003, background): “The following version of the Christ–Kiselev argument (see Christ and Kiselev (1998); Colliander and Kenig (2002); Smith and Sogge (2000)) will enable us to obtain ‘‘retarded estimates’’ from ‘‘nonretarded estimates.’’”
Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation (2007, methodology): “2) in [9] one can obtain that the estimate (2.”
The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line (2005, methodology): “In §6, we define the boundary forcing operator, adapted from [9], and prove the needed estimates for it.”
Regularity properties of the cubic nonlinear Schr\"odinger equation on the half line (2015, methodology): “The idea to use the restricted norm method in the case of IBVP with mild nonlinearities comes from [11].”
The Initial-Boundary Value Problem for the Korteweg–de Vries Equation (2005, methodology): “(b) is Lemma 5.5 in Colliander and Kenig (2002) and the proof of (c) is modelled on the proof of Lemma 5.5 in Colliander and Kenig (2002).”