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Existence and Uniqueness for Periodic Solutions of the Benjamin–Bona–Mahony Equation

1977; Society for Industrial and Applied Mathematics; Volume: 8; Issue: 5 Linguagem: Inglês

10.1137/0508062

1095-7154

L. A. Medeiros, G. Perla Menzala,

Nonlinear Photonic Systems

We consider the problem $u_t - u_{xx} + uu_x = 0$ in $ - \infty < x$, $t < \infty $ with initial data at $t = 0$ which is 1-periodic and the boundary condition $u(x + 1,t) = u(x,t)$ for all x, t; proving the existence and uniqueness of the solutions of such a problem. We use the semi-discrete approach together with the energy method.