An asymptotic nonlinear theory of the two superposed magnetic fluids is presented taking into account the spatial as well as temporal effects. A generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. The various stability criteria are derived from this equation. Obtained also are the bell shaped soliton and the ...

We consider 2-D Klein Gordon equation with quadratic nonlinearity and prove Strichartz type dispersive estimates for the global solution with small initial data in the Sobolev space H.

We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The bound state solutions are derived and the antiparticle bound state is discussed.

We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the u5-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.

We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional acti...