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We study the autowave front propagation in a medium with discontinuous characteristics and the conditions for its stabilization to a stationary solution with a large gradient at the interface between media in the one-dimensional case. The asymptotic method of differential inequalities, based on constructing an asymptotic approximation of the solution, is the main method of study. We develop an algorithm for constructing such an approximation for the solution of the moving front form in a medium with discontinuous characteristics. The application of such an algorithm requires a detailed analysis of the behavior of the solution in neighborhoods of two singular points: the front localization point and the medium discontinuity point. As a result, we obtain a system of equations for the front propagation speed; this distinguishes this paper from the previously published ones. The developed algorithm can be used to describe autowave propagation in layered media. The results can also be extended to the multidimensional case. [ABSTRACT FROM AUTHOR]

Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation with aggregation, nucleation, growth and breakage processes is solved analytically by use of the methods of scaling transformation group, observation and trial function. Symmetries, reduced equations, invariant solutions, exact solutions, existence of solutions, evolution analysis of dynamic behavior for solutions are presented. The exact solutions obtained can be compared with the numerical scheme. The obtained results also show that the method of scaling transformation group can be applied to study integro-partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional complex Ginzburg–Landau equation with Kerr law and power law nonlinearity. Using the conformable derivative approach and the bifurcation method, we effectively derived new explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and antikink wave solution, compacton) under different parameter conditions. The quasiperiodic, chaotic behavior and sensitivity analysis of the model is studied for different values of parameters after deploying an external periodic force. Finally, various 2D and 3D simulation figures are plotted to show the physical significance of these exact solutions. [ABSTRACT FROM AUTHOR]

The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In our previous work [38] , we investigate the global existence of strong solutions to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity. However, it is unknown whether the entropy remains its boundedness or not? Thus, the main novelty of this paper is to give a positive response to this problem. In fact, we show that the uniform boundedness of the entropy and the L 2 regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. Modified De Giorgi type iteration techniques and useful singularly weighted estimates are developed to deduce the lower and upper bounds on the entropy. [ABSTRACT FROM AUTHOR]

The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, which is a modified and generalized form of the ordinary KdV equation. We use the fractional operator in the Caputo sense to analyze the equation. We examine some theoretical results concerned with the solution's existence, uniqueness, and stability. We employ a modified Laplace method to extract the numerical results of the considered equation. We use MATLAB-2020 to simulate the results in a few fractional orders. We report the effects of the fractional order on the wave dynamics of the proposed equation. [ABSTRACT FROM AUTHOR]

Population balance equations are essential to scientists and engineers of widely varying discipline, applications have covered a wide range of processes involving bioreactors, catalytic reaction, and cell populations, but the lack of exact solutions necessitates the use of numerical methods. Group classification of Smoluchowski equation with a source is provided by use of the method of preliminary group classification. The determining equation is successfully solved by the method of developed Lie group analysis. All source functions, reduced equations, invariant solutions, explicit exact solutions and dynamic behavior analysis of solutions are presented in this paper. • Population balance equations for aggregation and breakage processes. • Explicit exact solutions of population balance equations with a source term. • Dynamic behavior analysis of solutions. • Preliminary group classification. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the kinetic description of the first-order Cucker-Smale (CS) flocking model on the real line. We reveal the equivalent relation between the measure-valued solution to the first- and second-order kinetic CS equations. The emergent behavior of the first-order kinetic CS equation and the characterization of the asymptotic solution are studied. We also provide the equivalent relation between classical/measure-valued solutions to the first-order kinetic CS equation and a classical solution to the second-order hydrodynamic CS equations, and present the corresponding analysis on the large-time behavior. The numerical experiments support our analysis and provide an efficient algorithm to obtain the asymptotic solution without simulating the model for a long time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the soliton molecules, asymmetric soliton and some novel hybrid solutions for the isospectral B-type Kadomtsev–Petviashvili (BKP) equation based on a new resonance condition. The soliton molecules and asymmetric soliton of the isospectral BKP equation can be obtained by selecting the appropriate parameters. Based on velocity resonance, module resonance and long-wave limit method, we can obtain the interactions of soliton molecules, breather waves and lump waves. Finally, we give the graphic of soliton molecules, asymmetric soliton and some novel hybrid solutions, and give the dynamic behavior analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the generation of interface for the solution of the mass conserved Allen-Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of ε, the solution generally develops a steep transition layer of thickness O(εγ) (0 < γ ≤ 1) at a certain time of order ε²|ln ε|. In some cases, we prove that the thickness of the interface is exactly of order \varepsilon, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen-Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate rational solitons and rogue wave (RW) solutions of the nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation by the generalized Darboux transformation method. The exact rational solutions are presented in terms of determinants whose entries are given by the initial eigenfunction and continuous-wave solution. Their dynamics investigated reveals the interesting patterns such as rational solitons, singular and nonsingular RWs depending on the selection of the free parameters. The compression effects of the RWs with respect to the strength of the higher-order nonlinear effects parameter γ are discussed and graphically illustrated. • Construction the rational soliton and higher-order RW solutions for the nonlocal LPD equation by the generalized DT. • Dynamic behavior analysis for the singular and nonsingular rational solutions. • The growing compressed effects of the nonsingular RW solutions are analyzed through numerical plots. [ABSTRACT FROM AUTHOR]

The Fokas-Lenells (FL) equation describes the propagation of ultra-short pulse in optical fiber. In this paper, optical soliton solutions are obtained through Hirota bilinear method, and the asymptotic behavior analysis and graphical simulation are given to reveal the propagation and dynamical behavior. It's worth noting that novel dynamic characteristics have been found with different choices of the wave numbers in the two-soliton and three-soliton solutions, which present different dynamical behavior from the general case. Moreover, some figures are given to better illustrate the dynamic characteristics. [ABSTRACT FROM AUTHOR]