Mário Figueira

EXISTENCE OF WEAK SOLUTIONS FOR A QUASILINEAR VERSION OF BENNEY EQUATIONS

Benney introduced a general strategy for deriving systems of nonlinear partial differential equations associated with long- and short-wave solutions. The semi-linear Benney system was studied recently by Tsutsumi and Hatano. Here, we tackle the nonlinear version of it and using compensated compactness techniques, we prove the global existence of weak solutions to the Cauchy problem, in the case that the equation for the amplitude of the long wave is a quasilinear conservation law with flux f(v) = av2 - bv3 where a, b are constants with b > 0.

On the existence of weak solutions for a nonlinear time dependent Dirac equation

In this paper we prove the existence of a weak solution of the Cauchy problem for the nonlinear Dirac equation in ℝ × ℝ where X ( r ) is the characteristic function of a compact interval of ]0, + ∞[

Supercritical Blowup in Coupled Parity-Time-Symmetric Nonlinear Schrödinger Equations

We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrodinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e., the ones describing self-action and nonlinear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonlinearity and the nonlinear coupling is repulsive, as well. The proofs are based on the virial technique arguments. Several particular cases are also illustrated numerically.

Finite-time blowup for a complex Ginzburg-Landau equation with linear driving

In this paper, we consider the complex Ginzburg–Landau equation

On the blowup of solutions of a Schrödinger equation with an inhomogeneous damping coefficient

{u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}

Spatial plane waves for the nonlinear Schrödinger equation: Local existence and stability results

ut=eiθ[Δu+|u|αu]+γu on

L^2

{\mathbb{R}^N}