Quasi self-adjoint nonlinear wave equations

Physical dynamics of quasi-particles in nonlinear wave equations

By treating the centers of solitons as point particles and studying their discrete dynamics, we demonstrate a new approach to the quantization of the soliton solutions of the sine-Gordon equation, one of the first model nonlinear field equations. In particular, we show that a linear superposition of the non-interacting shapes of two solitons offers a qualitative (and to a good approximation qua...

Self-adjoint differential-algebraic equations

Nonlinear Wave Equations

where := −∂2 t +∆ and u[0] := (u, ut)|t=0. The equation is semi-linear if F is a function only of u, (i.e. F = F (u)), and quasi-linear if F is also a function of the derivatives of u (i.e. F = F (u,Du), where D := (∂t,∇)). The goal is to use energy methods to prove local well-posedness for quasilinear equations with data (f, g) ∈ Hs × Hs−1 for large enough s, and then to derive Strichartz esti...

Nonlinear Wave Equations

This paper explores the properties of nonlinear wave equations. The proof for the existence and uniqueness of solutions to the 1+1 dimensional linear wave equation with smooth data is given. The D’Alembert formula is then presented in its full generality for the nonlinear equation. Important properties like the domain of dependence and propagation of information are discussed and motivated. The...

Nonlinear stochastic wave equations

In this paper we study the Cauchy problem for the semilinear stochastic wave equation (@ 2