Finite-Time Blow-Up for Solutions of Nonlinear Wave Equations.
Focusing and absorption of nonlinear oscillations
Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation
We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like , where stands for the wavelength....
Focusing of spherical nonlinear pulses in R1+3. II. Nonlinear caustic.
We study spherical pulse like families of solutions to semilinear wave equattions in space time of dimension 1+3 as the pulses focus at a point and emerge outgoing. We emphasize the scales for which the incoming and outgoing waves behave linearly but the nonlinearity has a strong effect at the focus. The focus crossing is described by a scattering operator for the semilinear equation, which broadens the pulses. The relative errors in our approximate solutions are small in the L∞ norm.
Forced oscillations of a class of nonlinear delay hyperbolic equations
Forced periodic vibrations of an elastic system with elastico-plastic damping
We prove the existence and find necessary and sufficient conditions for the uniqueness of the time-periodic solution to the equations for an arbitrary (sufficiently smooth) periodic right-hand side , where denotes the Laplace operator with respect to , and is the Ishlinskii hysteresis operator. For this equation describes e.g. the vibrations of an elastic membrane in an elastico-plastic medium.
Forced vibrations of wave equations with non-monotone nonlinearities
Fourier analysis of null forms and nonlinear wave equations.
Fourier integral operators and nonlinear wave equations
Free vibrations for the equation of a rectangular thin plate
In the paper, we deal with the equation of a rectangular thin plate with a simply supported boundary. The restoring force being an odd superlinear function of the vertical displacement, the existence of infinitely many nonzero time-periodic solutions is proved.
Front d'onde des fonctions non linéaires et polynômes