Integration of the Harry Dym equation with an integral type source

In the work, we deduce the evolution of scattering data for a spectral problem associated with the nonlinear evolutionary equation of Harry Dym with a self-consistent source of integral type. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Harry Dym equation with an integral type source.

Novel Soliton Molecules and Wave Interactions for A (3+1)-Dimensional Nonlinear Evolutionary Equation

New wave excitations are revealed for a (3+1)-dimensional nonlinear evolution equation to enrich nonlinear wave patterns in nonlinear systems. Based on a new variable separation solution with two arbitrary variable separated functions obtained by means of the multilinear variable separation approach, localized excitations of N dromions, N x M lump lattice and N x M ring soliton lattice are explored. Interestingly, it is observed that soliton molecules can be composed of diverse "atoms" such as the dromions, lumps and ring solitons, respectively. Elastic interactions between solitons and soliton molecules are graphically demonstrated.

The Evolution Equation for Wave Processes of the Shift Deformation

One-dimensional process of formation and the subsequent motion of a plane transverse shock wave is studied on the basis of solutions of the corresponding nonlinear evolutionary equation. This equation defines behaviour of the solution in front area of wave process and follows from inner series of a matched asymptotic expansions method. A comparative analysis of the volume deformation and forming processes will be carried out and their basic differences are specified. In the capacity of model examples solutions of some concrete boundary value problems of a dynamic shear deformation are observed.

The Global Existence of Nonlinear Evolutionary Equation with Small Delay

We investigate the global existence of the delayed nonlinear evolutionary equation∂tu+Au=f(u(t),u(t−τ)). Our work space is the fractional powers spaceXα. Under the fundamental theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.