Cauchy problem for the linearized KdV equation on general metric star graphs

Abstract

The Korteweg-de Vries (KdV) equation has attracted much attention in the literature, both in the context of physics and mathematics. This equation was found to permit soliton solutions and allow the modeling of solitary wave propagation on a water surface, a phenomena first discovered by Scott Russell in 1834. The KdV equation is also used, e.g., to model the unidirectional propagation of small amplitude long waves in nonlinear dispersive systems such as ion-acoustic waves in a collisionless plasma, and magnetosonic waves in a magnetized plasma etc [11]. The linearized KdV provides an asymptotic description of linear, undirectional, weakly dispersive long waves, for example, shallow water waves. Earlier, it was proven that via normal form transforms, the solution of the KdV equation can be reduced to the solution for the linear KdVequation [12]. Belashov and Vladimirov [12] numerically investigated the evolution of a single disturbance u(0, x) = u0 exp(−x/l) and showed that in the limit l → 0, u0l 2 = const, the solution of the KdV equation is qualitatively similar to the solution of the linearized KdV equation. Boundary value problems on half lines were considered in [2, 5, 7]. Here, summarizing and extending the results in [13], we investigate the linearized KdV equation on star graphs Γ with m + k semi-infinite bonds connected at one point, called the vertex. The bonds are denoted by Bj , j = 1, 2, ..., k + m, the coordinate xj on Bj is defined from −∞ to 0 for j = 1, 2, ..., k, and from 0 to +∞ for j = k + 1, ..., k +m such that on each bond, the vertex corresponds to 0. On each bond we consider the linear equation: ( ∂