Ion acoustic shock and periodic waves through Burgers equation in weakly and highly relativistic plasmas with nonextensivity

A comparative study is carried out for the nonlinear propagation of ion acoustic shock waves both for the weakly and highly relativistic plasmas consisting of relativistic ions and qdistributed electrons and positions.The Burgers equation is derived to reveal the physical phenomena using the well known reductive perturbation technique.The integration of the Burgers equation is performed by the（G￠/G）-expansion method.The effects of positron concentration,ion–electron temperature ratio,electron–positron temperature ratio,ion viscosity coefficient,relativistic streaming factor and the strength of the electron and positron nonextensivity on the nonlinear propagation of ion acoustic shock and periodic waves are presented graphically and the relevant physical explanations are provided.

Diversity of New Three-Wave Solutions and New Periodic Waves for the (3 + 1)-Dimensional Kadomtsev-Petviashvili-Boussinesq-Like Equation

Based on the generalized bilinear method, diversity of exact solutions of the (3 + 1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like equation is successfully derived by using symbolic computation with Maple. These new solutions, named three-wave solutions and periodic wave have greatly enriched the existing literature. Via the three-dimensional images, density images and contour plots, the physical characteristics of these waves are well described. The new three-wave solutions and periodic solitary wave solutions obtained in this paper, will have a wide range of applications in the fields of physics and mechanics.

Forces of fully nonlinear interfacial periodic waves on a cylindrical pile in a two-layer fluid with free-surface boundary conditions

In the frame of fully nonlinear potential flow theory,series solutions of interfacial periodic gravity waves in a two-layer fluid with free surface are obtained by the homotopy analysis method(HAM),and the related wave forces on a vertical cylinder are analyzed.The solution procedure of the HAM for the interfacial wave model with rigid upper surface is further developed to consider the free surface boundary.And forces of nonlinear interfacial periodic waves are estimated by both the classical and modified Mori-son equations.It is found that the estimated wave forces by the classical Morison equation are more conservative than those by the modified Morison’s formula,and the relative error between the total inertial forces calculated by these two kinds of Morison’s formulae remains over 25%for most cases unless the upper and lower layer depths are both large enough.It demonstrates that the convective acceleration neglected in the classical Morison equation is rather important for inertial force exerted by not only internal solitary waves but also interfacial periodic waves.All of these should further deepen our understanding of internal periodic wave forces on a vertical marine riser.

Solitons and periodic waves for a generalized(3+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics

Under investigation in this paper is a generalized(3+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics.Soliton and one-periodic-wave solutions are obtained via the Hirota bilinear method and Hirota-Riemann method.Magnitude and velocity of the one soliton are derived.Graphs are presented to discuss the solitons and one-periodic waves:the coefficients in the equation can determine the velocity components of the one soliton,but cannot alter the soliton magnitude;the interaction between the two solitons is elastic;the coefficients in the equation can influence the periods and velocities of the periodic waves.Relation between the one-soliton solution and one-periodic wave solution is investigated.

Periodic folded waves for a (2+1)-dimensional modified dispersive water wave equation

A general solution, including three arbitrary functions, is obtained for a （2~l）-dimensional modified dispersive water-wave （MDWW） equation by means of the WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and the degenerated single folded solitary waves are investigated graphically and found to be completely elastic.

Doubly periodic patterns of modulated hydrodynamic waves:exact solutions of the Davey-Stewartson system

Exact doubly periodic standing wave patterns of the Davey-Stewartson （DS） equations are derived in terms of rational expressions of elliptic functions.In fluid mechanics,DS equations govern the evolution of weakly nonlinear,free surface wave packets when long wavelength modulations in two mutually perpendicular,horizontal directions are incorporated.Elliptic functions with two different moduli （periods） are necessary in the two directions.The relation between the moduli and the wave numbers constitutes the dispersion relation of such waves.In the long wave limit,localized pulses are recovered.

Manipulating Backward Propagation of Acoustic Waves by a Periodical Structure

In the backward propagation of acoustic waves, the direction of phase velocity is anti-parallel to that of group velocity. We propose a scheme to manipulate the backward propagation using a periodicM structure. The dynamic backward propagation process is further experimentally observed. It is demonstrated that the oblique incident plane wave moves backward when it travels through the periodical structure and the backward shift can be controlled within a certain range.

Nonlinear Propagation of Positron-Acoustic Periodic Travelling Waves in a Magnetoplasma with Superthermal Electrons and Positrons

The nonlinear propagation of positron acoustic periodic （PAP） travelling waves in a magnetoplasma composed of dynamic cold positrons, superthermal kappa distributed hot positrons and electrons, and stationary positive ions is examined. The reductive perturbation technique is employed to derive a nonlinear Zakharov Kuznetsov equation that governs the essential features of nonlinear PAP travelling waves. Moreover, the bifurcation theory is used to investigate the propagation of nonlinear PAP periodic travelling wave solutions. It is found that kappa distributed hot positrons and electrons provide only the possibility of existence of nonlinear compressive PAP travelling waves. It is observed that the superthermality of hot positrons, the concentrations of superthermal electrons and positrons, the positron cyclotron frequency, the direction cosines of wave vector k along the z-axis, and the concentration of ions play pivotal roles in the nonlinear propagation of PAP travelling waves. Tile present investigation may be used to understand the formation of PAP structures in the space and laboratory plasmas with superthermal hot positrons and electrons.

NONLINEAR TRAVELING WAVES IN A COMPRESSIBLE MOONEY-RIVLIN ROD I.LONG FINITE-AMPLITUDE WAVES

In literature,nonlinear traveling waves in elastic circular rods have only been studied based on single partial differential equation(pde)models,and here we consider such a problem by using a more accurate coupled-pde model.We derive the Hamiltonian from the model equations for the long finite-amplitude wave approximation,analyze how the number of singular points of the system changes with the parameters,and study the features of these singular points qualitatively.Various physically acceptable nonlinear traveling waves are also discussed,and corresponding examples are given.In particular,we find that certain waves,which cannot be counted by the single-equation model,can arise.

Rogue waves for the(2+1)-dimensional Myrzakulov–Lakshmanan-Ⅳ equation on a periodic background

In this paper, the rogue wave solutions of the(2+1)-dimensional Myrzakulov–Lakshmanan(ML)-Ⅳ equation, which is described by five component nonlinear evolution equations, are studied on a periodic background. By using the Jacobian elliptic function expansion method, the Darboux transformation(DT) method and the nonlinearization of the Lax pair, two kinds of rogue wave solutions which are expressed by Jacobian elliptic functions dn and cn, are obtained.The relationship between these five kinds of potential is summarized systematically. Firstly, the periodic rogue wave solution of one potential is obtained, and then the periodic rogue wave solutions of the other four potentials are obtained directly. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Numerical Study of Interactions between Regular Water Waves and a Submerged Obstacle

The interaction between regular water waves and a submerged obstacle in a channel is studied numerically. The fluid viscosity is taken into account and the volume of fluid method is used to deal with the free surface. The incident regular waves are generated by use of numerical absorbing wave maker paddle. The present method can be used to predict the nonlinear deformations of the transmitted regular waves, and to simulate the vortex flow near the obstacle and the shear flows beneath the free surface.

Dynamical interactions between higher-order rogue waves and various forms ofn-soliton solutions(n→∞)of the(2+1)-dimensional ANNV equation

We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.

New exact periodic solutions to (2+1)-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.

ORBITAL STABILITY OF PERIODIC TRAVELING WAVE SOLUTIONS TO THE GENERALIZED ZAKHAROV EQUATIONS

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iu ＋uxx = uv ＋ ｜u｜^2u, vtt-vxx=（｜u｜^2）xx.First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].

Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation

In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.

A new type of internal solitary waves with a re-appearance period of 23 h observed in the South China Sea

The South China Sea （SCS）, in particular the northern SCS, is one of ocean areas where energetic internal solitary waves （ISWs）occur most frequently （Cai et al., 2012; Zheng, 2017）. Based on the re-appearance period （RP） at an observation station, Ramp et al.（2004） divided the ISWs into two types：Type-a and Type-b. Type-a ISWs arrive regularly at the same time every day, i.e., the RP is about 24 h, and Type-b ISWs arrive about one hour late every day, i.e., the RP is about 25 h.

Application of MEP Method to the Study of statistical Properties of Random Waves

The maximum entropy principle (MEP) method and the corresponding probability evaluation method are introduced, and the maximum entropy probability distribution expression is deduced in moment of the second order. Fully developed wave height distribution in deep water and wave height and period distribution for different depths in wind wave channel experiment are obtained from the MEP method, and the results are compared with the distribution and the experimental histogram. The wave height and period distribution for the Lianyungang port is also obtained by the MEP method, and the results are compared with the Weibull distribution and the field histogram.

SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS FOR THE NON-NEWTONIAN FILTRATION EQUATIONS WITH NONLINEAR SOURCES AND A TIME-VARYING DELAY

This paper is concerned with the non-Newtonian filtration equations with non- linear sources and a time-varying delay. By an extension of Mawhin＇s continuation theorem and some analysis methods, several sufficient conditions ensuring the existence of solitary wave and periodic wave solutions are obtained. Some corresponding results in the literature are improved and extended. An example is given to illustrate the effectiveness of our results.

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation

The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.

THE SMOOTH AND NONSMOOTH TRAVELLING WAVESOLUTIONS IN A NONLINEAR WAVE EQUATION

The travelling wave solutions (TWS) in a class of P. D. E. is studied. The travelling wave equation of this P. D. E. is a planar cubic polynomial system in three-parameter space. The study for M became the topological classifications of bifurcations of phase portraits defined by the planar system. B using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(xi), and considering discontinuity of the right side of the equation of TWS when xi = x - ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.