Bäcklund Transformation and Exact Solutions to a Generalized (3 + 1)-Dimensional Nonlinear Evolution Equation

In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, the N-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.

Degeneration of Solitons for a the (3+1)-dimensional Generalized Nonlinear Evolution Equation for the Shallow Water Waves

A the (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves is investigated with different methods. Based on symbolic computation and Hirota bilinear form, Nsoliton solutions are constructed. In the process of degeneration of N-soliton solutions, T-breathers are derived by taking complexication method. Then rogue waves will emerge during the degeneration of breathers by taking the parameter limit method. Through full degeneration of N-soliton, M-lump solutions are derived based on long wave limit approach. In addition, we also find out that the partial degeneration of N-soliton process can generate the hybrid solutions composed of soliton, breather and lump.

Nonlinear evolution equation for modeling waves at the boundary of a stratified flow of viscous liquids in an inclined channel

The dynamics of perturbations of the interface of a two-layer Poiseuille flow in a flat closed inclined channel is studied. The velocity profiles of wave motion are analytically found neglecting dissipation, dispersion and pumping of perturbations. On the basis of the found solution, a nonlinear evolution integro-differential equation for plane moderately long perturbations of the interface of the liquids is derived. The coefficients of the equation are represented by integrals over the layer thicknesses from functions depending on the stationary flow and perturbation profiles. The equation takes into account viscous dissipation: one of the integrals in this equation corresponds to dissipation in lion-stationary boundary layers, and the other corresponds to the transfer of energy from the flow to the wave. For the case of small flow velocities, the coefficients of the equation are analytically calculated. The equation has also been generalized to the quasi-two-dimensional case when the gradients along the transversal coordinate are small.

Lump Solutions, Multi Lump Solutions and More Soliton Solutions of a Novel (2+1)-dimensional Nonlinear Evolution Equation

Exact solutions of a new (2+1)-dimensional nonlinear evolution equation are studied. Through the Hirota bilinear method, the test function method and the improved tanh-coth and tah-cot method, with the assisstance of symbolic operations, one can obtain the lump solutions, multi lump solutions and more soliton solutions. Finally, by determining different parameters, we draw the three-dimensional plots and density plots at different times.

Progression of Aortic Dissection: An Application of Soliton Solutions of Nonlinear Evolution Equation in (3+1)-Dimensions

Aortic dissection is a serious pathology involving the vessel wall of the aorta with significant societal impact. To understand aortic dissection we explain the role of the dynamic pathology in the absence or presence of structural and/or functional abnormalities. We frame a differential equation to evaluate the impact of mean blood pressure on the aortic wall and prove the existence and uniqueness of its solution for homeostatic recoil and relaxation for infinitesimal aortic tissue. We model and analyze generalized (3+1)-dimensional nonlinear partial differential equation for aortic wave dynamics. We use the Lie group of transformations on this nonlinear evolution equation to obtain invariant solutions, traveling wave solutions including solitons. We find that abnormalities in the dynamic pathology of aortic dissection act as triggers for the progression of disease in early-stage through the formation of soliton-like pulses and their interaction. We address the role of unstable wavefields in waveform dynamics when waves are unidirectional. Moreover, the notion of dynamic pathology within the domain of vascular geometry may explain the evolution of aneurysms in cerebral arteries and cardiomyopathies even in the absence of anatomical and physiological abnormalities.