This work deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues ...

We consider solutions to a nonlinear reaction diffusion equation when the reaction term varies randomly with respect to the spatial coordinate. The nonlinearity is either the ignition nonlinearity or the bistable nonlinearity, under suitable restrictions on the size of the spatial fluctuations. It is known that the solution develops an interface which propagates with a well-defined speed in the...

We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high-intensity ultrasound propagation as widely used in medical imaging and therapy. The usual term the standard model using Westervelt pressure formulation is of form ppt. However, this should be considered low-order approximation to more complex physical where higher order terms wi...