Smooth solutions to a class of quasilinear wave equations

Global solutions of quasilinear wave equations

has a global solution for all t ≥ 0 if initial data are sufficiently small. Here the curved wave operator is ̃g = g ∂α∂β, where we used the convention that repeated upper and lower indices are summed over α, β = 0, 1, 2, 3, and ∂0 = ∂/∂t, ∂i = ∂/∂x i, i = 1, 2, 3. We assume that gαβ(φ) are smooth functions of φ such that gαβ(0)= mαβ , where m00=−1, m11= m22= m33=1 and mαβ= 0, if α 6=β. The resul...

Modulating pulse solutions for quasilinear wave equations

This paper presents an existence proof for symmetric modulating pulse solutions of a quasilinear wave equation. Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave-train; they are also referred to as ‘moving breathers’ since they are time-periodic in a moving frame of reference. The problem is formulated as an infinite-...

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

Existence and Uniqueness of Solutions of Quasilinear Wave Equations (ii)

In this work we prove a result concernig the existence and uniqueness of solutions of quasilinear wave equation and we consider also their trivial solutions. We consider the following initial-boundary value problem for the nonlinear wave equation in the form u+ f (u) + g (u̇) = 0 in [0, T ) × Ω (QL) with initial values u0 = u (0, ·) , u1 = u̇ (0, ·) and boundary vale null, that is, u (t, x) = 0 o...

Periodic Wave Shock solutions of Burgers equations

In this paper we investigate the exact peroidic wave shock solutions of the Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the cauchy problem for viscid equation with small parametr ε and to discuss in particular the case of periodic wave shock. We show that the solution of this problem approaches the shock type solution for the cauchy problem of the in...