The Korteweg-de Vries equation on the quarter plane with asymptotically t-periodic data via the Fokas method

Article type: Research Article

Authors: Moon, Byungsoo^a | Hwang, Guenbo^{b; *}

Affiliations: [a] Department of Mathematics, Incheon National University, Incheon 22012, Korea. E-mail: [email protected] | [b] Department of Mathematics and Institute of Basic Science, Daegu University, Gyeongsan Gyeongbuk 38453, Korea. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We study the Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic boundary data for large t>0. We derive an expression for the Dirichlet to Neumann map to all orders in the perturbative expansion of a small ϵ>0 in the case of the asymptotically periodic boundary data. More precisely, we show that if the unknown Neumann boundary data are asymptotically periodic for large t in the sense that ux(0,t) and uxx(0,t) tend to periodic functions g˜1(t) and g˜2(t) for large t, respectively, then the periodic functions g˜1(t) and g˜2(t) can be characterized in terms of the given asymptotically periodic Dirichlet boundary datum u(0,t). Moreover, we determine effectively the Fourier coefficients of the functions g˜1(t) and g˜2(t) by solving a certain recursive algebraic equations.

Keywords: Initial-boundary value problem, integrable systems, Korteweg-de Vries equation, Dirichlet to Neumann map

DOI: 10.3233/ASY-171452

Journal: Asymptotic Analysis, vol. 107, no. 3-4, pp. 115-133, 2018