Lump,soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [1] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation’s dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas.

     

The influence of normal flow boundary conditions on spurious modes in finite element solutions to the shallow water equations

Finite element solutions of the primitive equation (PE) form of the shallow water equations are notorious for the severe spurious 2Δx modes which appear. Wave equation (WE) solutions do not exhibit these numerical modes. In this paper we show that the severe spurious modes in PE solutions are strongly influenced by essential normal flow boundary conditions in the coupled continuity-momentum system of equations. This is demonstrated through numerical examples that avoid the use of essential normal flow boundary conditions either by specifying elevation values over the entire boundary or by implementing natural flow boundary conditions in the weak weighted residual form of the continuity equation. Results from a series of convergence tests show that PE solutions are of nearly the same quality as WE solutions when spurious modes are suppressed by alternative specification of the boundary conditions. Network intercomparisons indicate that varying nodal support does not excite spurious modes in a solution, although it does enhance the spurious modes introduced when an essential normal flow boundary condition is used. Dispersion analysis of discrete equations for interior and boundary nodes offers an explanation of the observed solution behaviour. For certain PE algorithms a mixed situation can arise where the boundary nodes exhibit a monotonic (noise-free) dispersion relationship and the interior nodes exhibit a folded (noisy) dispersion relationship. We have found that the mixed situation occurs when all boundary nodes are specified elevation nodes (which are enforced as essential conditions in the continuity equation) or when specified flow boundary nodes are treated as natural boundary conditions in the continuity equation. In either case the effect is to generate a solution that is essentially free of noise. Apparently, the monotonic dispersion behaviour at the boundaries suppresses the otherwise noisy behaviour caused by the folded dispersion relation on the interior.     

Piezoelectromagnetic waves in a ceramic plate between two ceramic half-spaces

We analyze the propagation of piezoelectromagnetic waves guided by a plate of polarized ceramics between two ceramic half-spaces. An exact dispersion relation is obtained, which reduces to a few known elastic, electromagnetic, and quasistatic piezoelectric wave solutions in the literature as special cases. Numerical solutions to the equation that determines the dispersion relation show the existence of guided waves. The results are useful for acoustic wave and microwave devices.     

非圆截面杆中的非线性扭转波及其行波解

研究了非圆截面杆中非线性扭转波的传播特性.由于非圆截面杆的扭转运动会伴随有横截面的翘曲,这种翘曲运动将引起扭转波的弥散.如果同时考虑有限扭转变形和翘曲弥散的共同作用,将会得到非线性扭转波的方程.在相平面上,对非线性扭转波动方程进行定性分析,结果表明,在一定条件下方程存在同宿轨道或异宿轨道,分别相应于方程的孤波解或冲击波解.本文利用Jacobi椭圆函数展开法,对该非线性方程进行求解,得到了非线性波动方程的三类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.这些条件与定性分析的结果相一致.     

Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws

This paper analyzes the non-classical shock waves which arise as limits of certain diffusive-dispersive approximations to hyperbolic conservation laws. Such shocks occur for non-convex fluxes and connect regions of different convexity. They have negative entropy dissipation for a single convex entropy function, but not all convex entropies, and do not obey the classical Oleinik entropy criterion. We derive necessary conditions for the existence of non-classical shock waves, and construct them as limits of traveling-wave solutions for several diffusive-dispersive approximations. We introduce a “kinetic relation” to act as a selection principle for choosing a unique non-classical solution to the Riemann problem. The convergence to non-classical weak solutions for the Cauchy problem is investigated. Using numerical experiments, we demonstrate that, for the cubic flux-function, the Beam-Warming scheme produces non-classical shocks while no such shocks are observed with the Lax-Wendroff scheme. All of these results depend crucially on the sign of the dispersion coefficient. (Accepted February 8, 1996)     

Localization of Shear Waves Near Layers Separating Two Regularly Laminated Half-Spaces

The existence conditions for surface and normal shear waves in finite and infinite periodically laminated structures with broken translational symmetry are studied theoretically and numerically. The problems posed are reduced to systems of linear algebraic equations. The existence condition for their nontrivial solutions yield dispersion relations. Conditions for the existence of surface and normal shear waves are established. Some results are plotted. The dependence of the dispersion spectrum on the physical and geometrical properties of the symmetry breaker is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 128–134, January 2005.     

Controllable behaviors of nonautonomous solitons on background of continuous wave and cnoidal wave in $$\mathcal {}$$-symmetric dimer with inhomogeneous effect

We obtain exact \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric nonautonomous soliton solutions on background waves. These solutions indicate that dispersion and nonlinear coefficients influence form factors of nonautonomous solitons such as amplitude, width and center; however, linear coupling coefficient and gain/loss parameter only influence phase of solitons. Based on these solutions, the controllable behaviors such as postpone, sustainment and restraint on continuous wave background in an exponential decreasing dispersion system are discussed. Moreover, the propagation behaviors of solitons on the cnoidal wave background in different dispersion systems are also studied.     

Analysis of One-Dimensional Advection–Diffusion Model with Variable Coefficients Describing Solute Transport in a Porous medium

This work presents the analytical solution and temporal moments of one-dimensional advection–diffusion model with variable coefficients. Two case studies along with the two different sets of boundary conditions are considered at the inlet and outlet of the domain. In the first case, a time-dependent solute dispersion in the homogeneous domain along uniform flow is taken into account, whereas in the second case, due to inhomogeneity of domain, velocity is taken spatially dependent and the dispersion is assumed proportional to the square of the velocity. The Laplace transform is used to obtain the analytical solutions. The analytical temporal moments are derived from the Laplace domain solutions. To verify the correctness of the analytical solutions, a high-resolution second-order finite volume scheme is applied. Different case studies are considered and discussed. Both analytical and numerical results are in good agreement with each other.     

Harmonic waves in elastic sandwich plates

Motions of a sandwich plate with symmetric facings are studied in the framework of the three-dimensional equations of elasticity. Both the core and facings are assumed to be isotropic and linearly elastic.Harmonic wave solutions, which satisfy traction-free face conditions and continuity conditions of tractions and displacements at the interfaces, are obtained for four cases: symmetric plane strain solutions for extensional motion, antisymmetric plane strain solutions for flexural motion, and solutions for the symmetric and antisymmetric SH-waves. The dispersion relation for each of these cases is obtained and computed. In order to exhibit the effect of the ratios of facing to core thicknesses, elastic stiffnesses and densities, on the dynamic behavior of sandwich plates, dispersion curves are computed and compared for plates with thick, light, and soft facings as well as for plates with thin, heavy, and stiff facings. Asymptotic expressions of dispersion relations for extensional, flexural, and symmetric SH-waves are obtained in explicit form, as the frequencies and wave numbers approach zero.The thickness vibrations in sandwich plates are studied in detail. The resonance frequencies and modal functions of the thickness-shear and thickness-stretch motions are obtained. Simple algebraic formulas for predicting the lowest thickness-shear and the lowest thickness-stretch frequencies are deduced. The orthogonality of the thickness modal functions is established.     

Dispersion analysis of the least‐squares finite‐element shallow‐water system

The frequency or dispersion relation for the least‐squares mixed formulation of the shallow‐water equations is analysed. We consider the use of different approximation spaces corresponding to co‐located and staggered meshes, respectively. The study includes the effect of Coriolis, and the dispersion properties are compared analytically and graphically with those of the mixed Galerkin formulation. Numerical solutions of a test problem to simulate slow Rossby modes illustrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Asymptotic Stability of N-Solitary Waves of the FPU Lattices

We study stability of N-solitary wave solutions of the Fermi-Pasta-Ulam (FPU) lattice equation. Solitary wave solutions of the FPU lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable. In place of standard variational arguments for Hamiltonian systems, we use an exponential stability property of the linearized FPU equation in a weighted space which is biased in the direction of motion. The dispersion of the linearized FPU equation balances the potential term for low frequencies, whereas the dispersion is superior for high frequencies.We approximate the low frequency part of a solution of the linearized FPU equation by a solution to the linearized Korteweg-de Vries (KdV) equation around an N-soliton solution. We prove an exponential stability property of the linearized KdV equation around N-solitons by using the linearized Bäcklund transformation and use the result to analyze the linearized FPU equation.     

Laboratory Research Results of High-Density Solution Migration in Fresh Groundwaters

The mechanism of migration of high-density solutions injected into fresh water was studied in the laboratory by physical models. Fifty-four laboratory tests were performed using a sand box constructed of transparent plastic. This physical model represented a portion of the groundwater flow. A high-density flow was simulated using solutions of various densities and chemical composition. It was found that migration of high-Density solutions in ground water is in many instances governed by the relation between the density of high-density solution and that of the ground water. Peculiarities of temporal and spatial contaminant distribution in the dispersion halo, the effect of filtration flow velocity, the relationship of flowrates between the fresh-water and contaminant flows, and the impact of the model boundaries and gravity were determined. Dependence of the dispersion haloes shape upon the structure of the fresh-water flow is described. The paper examines migration of low-density solutions over high-density solutions and the behavior of high-density solutions under conditions of discharge at the surface. The results show that the migration of high-density solutions is distinctly three-dimensional, and its prediction is possible only when based on three-dimensional numerical models.     

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.     

Analytical Solutions for Solute Transport in a Spherically Symmetric Divergent Flow Field

Exact analytical solutions for an equation describing advection, dispersion, first-order decay, and rate-limited sorption of a solute in a steady, hemispherical or spherically symmetric, divergent flow field are presented for constant concentration and constant flux boundary conditions in a porous medium. The partial differential equation describing transport is a confluent hypergeometric equation that may be solved with variable substitution and Laplace transform, and the solutions are expressed by parabolic cylindrical functions. The novel solutions derived here may be applied to predict concentration distributions in space and time for porous media transport in a spherically symmetric flow field or for the special case where injection is just below a confining layer (hemispherical flow). The analytical solutions can be used to simulate wastewater injection from short-screened wells into thick formations or to analyze tracer tests that use short-screened wells to create approximately spherical flow fields in thick formations.     

Dispersion Waves of Two Fluids in a Porous Medium

The dispersion of two fluids in a porous medium is analyzed as a wave process. The wave equations are derived, and for plane wave solutions a wave number versus frequency dispersion relation is obtained. Suitable choices for the saturation dependence of terms in the equations of motion and the dynamic pressure difference equation lead to physical solutions.     

Weak Solutions of a Generalized Boussinesq System

We study the convergence of homoclinic orbits and heteroclinic orbits in the dynamical system governing traveling wave solutions of a perturbed Boussinesq systems modeling two-directional propagation of water waves. Nonanalytic weak solutions are found to be limits of these orbits, including compactons, peakons, and rampons, as well as infinitely many mesaons occurring at the same fixed point in the dynamical system. Singularities of solitary wave solutions in the system are also studied to understand the important impact of both linear and nonlinear dispersion terms on the regularity of these solutions.     

Experiments on Solute Transport in Aggregated Porous Media: Are Diffusions Within Aggregates and Hydrodynamic Dispersion Independent?

Two region models for solute transport in porous media assume that hydrodynamic dispersion in mobile water and solute diffusion within immobile water regions are independent. Experimental and theoretical results for transport through a macropore indicate that hydrodynamic dispersion and solute exchange are interdependent. Experiments were carried out to investigate this problem for a column packed with spherical porous aggregates. The effective diffusion coefficient of a tracer within the agreggates was determined from specific experiments. The dispersivity of the bed was determined from experiments carried out with a column filled with nonporous beads. We took advantage of the dependence of hydrodynamic dispersion on density ratios between the invading and displaced solutions to obtain a set of breakthrough curves corresponding to situations where the diffusion coefficient remains constant, whereas the dispersivity varies. Simulations reproduce correctly the experiments. Small discrepancies are noted that can be corrected either by increasing the dispersion coefficient or by fitting the external mass transfer coefficient. Increased dispersion coefficients probably reveal a modification of Taylor dispersion due to solute exchange. The fitted external mass transfer coefficients are close to the values obtained with classical correlations of the chemical engineering literature.     

Surface and interfacial waves in ionic crystals

The continuum linear theory of ionic crystals is applied to develop a two-dimensional eigenvalue problem in the Stroh formalism. An integral approach is exploited to study the occurrence of surface waves along a free boundary of the crystal. Dispersion relations are obtained by separating real and imaginary parts of the governing system and various boundary conditions are examined. The problem of interfacial waves along the separation boundary between two different crystals is also outlined. Numerical computations are performed for a centrosymmetric crystal (KCl) in order to evaluate bulk wave speeds, limiting speed of surface waves and solutions to the dispersion equations for different boundary conditions.     

Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation

In this paper, we consider an extended nonlinear Schrödinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.