Initial-Boundary Value Problems for Nonlinear Dispersive Equations of Higher Orders Posed on Bounded Intervals with General Boundary Conditions

The present article concerns general mixed problems for nonlinear dispersive equations of any odd-orders posed on bounded intervals. The results on existence, uniqueness and exponential decay of solutions are presented.

Scaling Symmetries and Conservation Laws for Variable-coefficients Nonlinear Dispersive Equations

Scaling symmetries arise in different branches of physics, and symmetry-based approaches are powerful tools for studying scaling-invariant models since they can provide conservation laws that are not obvious by inspection. In this framework, the class of variable-coefficients nonlinear dispersive equations vc$K(m,n)$, which contains several important evolution equations modeling nonlinear phenomena, is considered. For some of its scaling-invariant subclasses, we study its nonlinear self-adjointness and construct eight new local conservation laws associated with scaling symmetries by using a general theorem on conservation laws and the multipliers method. The property of scale invariance of those equations led to five conservation laws with a direct physical interpretation: energy, center of mass, and mass are the conserved quantities obtained in some cases.