In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of two models, namely standing wave NLS traveling waves, which turn out to have compact support, compactons. show that compactons are unique bell-shaped corresponding PDEs for appropriate variational problems as well. provide a complete spect...

As a sequel of the recent works, we would like to discuss another stage for the solution of the mixed problem which considers the concepts of well operated conditions applicable to the solution of mixed problem, i.e. the existence and uniqueness of the solution must always conformable with some assumptions.

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

Zero curvature formulations are given for the “dual hierarchies” of standard soliton equation hierarchies, recently introduced by Olver and Rosenau, including the physically interesting Fuchssteiner-Fokas-CamassaHolm hierarchy. PACS Numbers: 03.40.Kf, 47.20.Ky, 52.35.Sb Through the years since the discovery of the notion of “integrability” in PDEs, quite a number of integrable PDEs have been di...