Integrable Nonlinear Evolution Equations on the Half-Line

By A. S. Fokas. 2002

300 citations recorded by the source corpus.

Subfields: Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5de35260bf0e1001d17e2961ccffa7a9f3ae96e5

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In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

The generalized Dirichlet-Neumann map for linear elliptic PDEs and its numerical implementation (2008, unknown): “This includes linear evolution PDEs with spatial derivatives of arbitrary order on the half-line [7] and on a finite interval [10], the Laplace, the bi-harmonic and the modified Helmholtz equation in certain simple polygons [1–4] and the basic nonlinear integrable evolution PDEs on the half-line for”
Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval (2018, methodology): “In 1997, Fokas [7] (also see [8, 9]) introduced a new method, the so-called Fokas method, for analyzing boundary value problems for linear and for integrable nonlinear PDEs.”
The Degasperis-Procesi equation on the half-line (2012, methodology): “), g 2(t) dened by u 0(x) = u(x;0); 0 <x<1; g 0(t) = u(0;t); g 1(t) = u x(0;t); g 2(t) = u xx(0;t); 0 <t<T: (1.3) The main peculiarity compared with other applications of the approach of [15, 16] is that the Lax pair involves 3 3-matrices instead of 2 2-matrices. This dierence leads to s”
The unified method: I. Nonlinearizable problems on the half-line (2011, methodology): “This method has been implemented to linear and integrable nonlinear evolutions PDEs on the half-line and the finite interval [6, 8, 9, 13, 16, 24, 25, 31, 32, 34, 36, 39, 44, 45, 49, 50, 56, 57, 61, 62, 64, 71, 72, 73, 75, 76, 79, 84] to linear and integrable nonlinear evolution PDEs in two space va”
The generalized Dirichlet‐to‐Neumann map for certain nonlinear evolution PDEs (2005, methodology): “It was shown in [12, 17] that there exist particular choices of boundary conditions for which the global relation can be solved using algebraic manipulations without the need to formulate a nonlinear Volterra integral equation.”
Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line (2003, background): “The solution of this initial boundary value (IBV) problem can be constructed as follows ([3, 6]): Letters in Mathematical Physics 65: 199–212, 2003.”
Regularity properties of the cubic nonlinear Schr\"odinger equation on the half line (2015, methodology): “Finally, we should also mention that in the integrable case (p = 4) Fokas, [17], obtained a solution of (1) when g is a Schwartz function and h is sufficiently smooth by reformulating the problem as a 2 × 2 matrix Riemann-Hilbert problem.”
The Initial-Boundary Value Problem for the Korteweg–de Vries Equation (2005, methodology): “A global well-posedness result for the problem (1.1)(1.8) is obtained by [Fam04] for s 0. Inverse scattering techniques have been applied to the problem (1.2) by [ Fok02 ] and the linear analogue of the problem (1.5) in [FP01] for Schwartz class data.”
THE mKdV EQUATION ON THE HALF-LINE (2004, methodology): “To describe this class, we will follow the approach proposed in [7,9].”
The nonlinear Schrödinger equation on the half-line (2004, unknown): “This involves only tedious but straightforward algebraic manipulations (see [1] for details).”
The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line (2005, background): “[10] in the integrable case α = 3, λ = ±2 with φ Schwartz and f sufficiently smooth, obtained a solution to (1.”