Existence and stability of traveling wave solutions to one-sided mixed initial-boundary value problem for first-order quasilinear hyperbolic systems

Abstract

When one characteristic of the system is linearly degenerate, under suitable boundary conditions, we get the existence of traveling wave solutions located on the corresponding characteristic trajectory to the one-sided mixed initial-boundary value problem. When the system is linearly degenerate, by introducing the semi-global normalized coordinates, we derive the related formulas of wave decomposition to prove the stability of traveling wave solutions corresponding to all leftward and the rightmost characteristic trajectories. Finally, for the traveling wave solutions corresponding to other rightward characteristic trajectories, some examples show their possible instability. Recently, the authors of this paper have given the specific form of n families of C 1 traveling wave solutions to the Cauchy problem (1.1) and (1.11) under the LD hypothesis and proved the stability of the leftmost and rightmost families of traveling wave solutions. Moreover, it is also shown that the intermediate families of traveling wave solutions may lose their stability [22]. In order to study the effect of boundary conditions on the stability of traveling wave solutions, in this paper, we consider the one-sided mixed initial-boundary value problem for system (1.1) on the domain f.t, x/ j t 0, x 0g with the initial-boundary condition:t D If there exist a vector function U.x/ 2 C 1 .R/ and a constant 2 R, such that U.x t/ is a C 1 solution to quasilinear hyperbolic system (1.1) with the boundary condition (1.12), then u D U.x t/ is called to be a traveling wave solution.For any given y 2 R n and i 2 N , consider the ith characteristic trajectory u D u .i/ . ; y/ passing through u D y in the u-space, namely ( du .i/ . ;y/ ds D r i .u .i/ . ; y//, D 0 : u .i/ .0; y/ D y.(1.14)According to the theory of ordinary differential equations, by (1.7), the initial value problem (1.14) admits a global C 1 solution u D u .i/ . ; y/ for 2 R.

Stability of traveling wave solutions to the initial–boundary value problem for diagonalizable quasilinear hyperbolic systems

“…First of all, we give the following two lemmas describing the interactions between different waves (see [1,11]), which play an important role on the proof of main result.…”

“…recently, in [1] the authors proved the stability of all the negative eigenvalues family and the rightmost one of traveling wave solutions and possible instability of the intermediate and positive families of traveling wave solutions of the mixed initial-boundary value problem for system (1.1) with small initial data. In this article, we will investigate the existence and stability of all families traveling wave solutions to the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic system with 'large' initial data.…”