Periodic,breather and rogue wave solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid dynamics

Under investigation in this paper is a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation, which describes the propagation of nonlinear waves in fluid dynamics. Periodic wave solutions are constructed by virtue of the Hirota–Riemann method. Based on the extended homoclinic test approach, breather and rogue wave solutions are obtained. Moreover, through the symbolic computation, the relationship between the one-periodic wave solutions and one-soliton solutions has been analytically discussed, and it is shown that the one-periodic wave solutions approach the one-soliton solutions when the amplitude $\eta \to 0$.     

Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions.As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed.     

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Nonlinear convection–diffusion equations with nonlocal flux and possibly degenerate diffusion arise in various contexts including interacting gases, porous media flows, and collective behavior in biology. Their numerical solution by an explicit finite difference method is costly due to the necessity of discretizing a local spatial convolution for each evaluation of the convective numerical flux, and due to the disadvantageous Courant–Friedrichs–Lewy (CFL) condition incurred by the diffusion term. Based on explicit schemes for such models devised in the study of Carrillo et al. a second‐order implicit–explicit Runge–Kutta (IMEX‐RK) method can be formulated. This method avoids the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but entails the necessity to solve nonlinear algebraic systems in every time step. It is proven that this method is well defined. Numerical experiments illustrate that for fine discretizations it is more efficient in terms of reduction of error versus central processing unit time than the original explicit method. One of the test cases is given by a strongly degenerate parabolic, nonlocal equation modeling aggregation in study of Betancourt et al. This model can be transformed to a local partial differential equation that can be solved numerically easily to generate a reference solution for the IMEX‐RK method, but is limited to one space dimension.     

Painlevé analysis,lump-kink solutions and localized excitation solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated. The integrability test is performed yielding a positive result. Through the Painlevé–Bäcklund transformation, we derive four types of lump-kink solutions composed of two quadratic functions and N exponential functions. It is shown that fission and fusion interactions occur in the lump-kink solutions. Furthermore, a new variable separation solution with two arbitrary functions is obtained, the localized excitations including lumps, dromions and periodic waves are analyzed by some graphs.     

Existence of homoclinic solutions for nonlinear second-order coupled systems

This work presents sufficient conditions for the existence of homoclinic solutions for second order coupled discontinuous systems of differential equations on the real line without the usual growth condition in the literature.The arguments apply the fixed point theory, Green's functions technique, $L^1$-Carathéodory functions, lower and upper solutions and Schauder's fixed point theorem.