New Exact Solutions to Long-Short Wave Interaction Equations

New exact solutions expressed by the Jacobi elliptic functions are obtained to the long-short wave interaction equations by using the modified F-expansion method. In the limit case, solitary wave solutions and triangular periodic wave solutions are obtained as well.

New Lump Solution and Their Interactions with N-Solitons for a Shallow Water Wave Equation

By employing the Hirota’s bilinear method and different test functions, the breather solutions of HSI equation with different structures are obtained based on symbolic calculation with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions. The interaction between lump solution and soliton solution is constructed in the form of lump solution, and the motion trajectory of lump is obtained. In addition, the theorem of lump solitons and N-solitons superposition is given and proved. The superposition formula of lump is derived from the theorem, and its spatial evolution behavior is given.

Interaction of elementary waves for relativistic Euler equations

In this paper, using the characteristic analysis method, we study the relativistic Euler equations of conservation laws in energy and momentum in special relativity. The interactions of elementary waves for the relativistic Euler equations are shown. The collision of two shocks, two centered rarefaction waves, a shock and a rarefaction wave yield corresponding ransmitted waves. The overtaking of two shocks appears a transmitted shock wave, together with a reflected centered rarefaction wave.

Non-completely elastic interactions in a(2+1)-dimensional dispersive long wave equation

With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the （24-1）-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.

New Periodic Wave Solutions and Their Interaction for （2＋1）-dimensional KdV Equation

A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contains two arbitrary functions） got by means of multilinear variable separation approach for （2＋1）-dimensional KdV equation. Limiting cases are considered and some localized excitations are derived, such as dromion, multidromions, dromion-antidromion, multidromions-antidromions, and so on. Some solutions of the dromion-antidromion and multidromions-antidromions are periodic in one direction but localized in the other direction. The interaction properties of these solutions, which are numerically studied, reveal that some of them are nonelastic and some are completely elastic. Furthermore, these results are visualized.

Global Well-posedness of the Generalized Long-short Wave Equations

In the present paper, we investigate the well-posedness of the global solutionfor the Cauchy problem of generalized long-short wave equations. Applying Kato＇s methodfor abstract quasi-linear evolution equations and a priori estimates of solution,we get theexistence of globally smooth solution.

Applying the New Extended Direct Algebraic Method to Solve the Equation of Obliquely Interacting Waves in Shallow Waters

In this study,the potential Kadomtsev-Petviashvili(pKP)equation,which describes the oblique interaction of surface waves in shallow waters,is solved by the new extended direct algebraic method.The results of the study show that by applying the new direct algebraic method to the pKP equation,the behavior of the obliquely interacting surface waves in two dimensions can be analyzed.This article fairly clarifies the behaviors of surface waves in shallow waters.In the literature,several mathematical models have been developed in attempt to study these behaviors,with nonlinear mathematics being one of the most important steps;however,the investigations are still at a level that can be called‘baby steps’.Therefore,every study to be carried out in this context is of great importance.Thus,this study will serve as a reference to guide scientists working in this field.

Lump and interaction solutions to the (3+1)-dimensional Burgers equation

The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.

STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

Numerical study of edge waves using extended Boussinesq equations

An edge wave numerical model was developed based on extended Boussinesq equations with the internal wave-generation method. The form of edge waves near a seawall was chosen as the input signal in order to avoid treatment of the moving shoreline on a sloping beach. As there was an energy transfer between different edge wave modes, not only the target mode but also other modes appeared in the simulations. Due to the nonlinear effect, the simulation results for mode-0 edge waves were slightly modulated by mode-1 and mode-2 waves. As the magnitudes of these higher-mode waves are not significantly related to those of the target mode, the internal wave-generation method in Boussinesq equations can produce high-quality edge waves. The numerical model was used to investigate the nonlinear properties of standing edge waves, and the numerical results were in strong agreement with theory.

GENERAL SOLUTION FOR INTERACTION OF SOLITARY WAVES INCLUDING HEAD-ON COLLISIONS

A general solution of the Boussinesq equation is presented which solves the problem of interaction of any number of right-going and left-going solitary waves.The solution relies on the exact solu- tion of Gardner,Greene,Kruskal,and Miura(1967),and has the same degree of accuracy as that solution, but has a wider scope of application.It is much simpler than,but as accurate as,Hirota's exact solu- tion(1973)of the Boussinesq equation,to which the present solution is compared for the simplest case of two solitary waves in head-on collision.

Generalized Mean-Flow Theory of Wave-Current-BottomInteractions

The interaction between waves, currents and bottoms in estuarine and coastal regions is ubiquitious, in particular the dynamic mechanism of waves on large-scale slowly varying currents. The wave action concept may be extended and applicated to the study of the mechanism. Considering the effects of moving bottoms and starting from the Navier-Stokes equation of motion of a vinous fluid including the Coriolis force, a generalized mean-flow medel theory for the nearshore region, that is, a set of mean-flow equations and their generalized wave action equation involving the three new kinds of actions termed respectively as the current wave action, the bottom wave action and the dissipative wave action which can be applied to arbitrary depth over moving bottoms and ambient currents with a typical vertical structure, is developed by vertical integration and time-averaglng over a wave peried, thus extending the classical concept, wave action, from the ideal averaged flow conservative system to the real averaged flow dissipative dynamical system, and having a large range of application.

THE INTERACTIONS BETWEEN WAVE-CURRENTS AND OFFSHORE STRUCTURES WITH CONSIDERATION OF FLUID VISCOSITY

Study of the how held around the large scale offshore structures under the action of waves and viscous currents is of primary importance for the scouring estimation and protection in the vicinity of the structures. But very little has been known in its mechanism when the viscous effects is taken into consideration. As a part of the efforts to tackle the problem, a numerical model is presented for the simulation of the how held around a fixed vertical truncated circular cylinder subjected to waves and viscous currents based on the depth-averaged Reynolds equations and depth-averaged k-epsilon turbulence model. Finite difference method with a suitable iteration defect correct method and an artificial open boundary condition are adopted in the numerical process. Numerical results presented relate to the interactions of a pure incident viscous current with Reynolds number Re = 10(5), a pure incident regular sinusoidal wave, and the coexisting of viscous current and wave with a circular cylinder, respectively. Flow fields associated with the hydrodynamic coefficients of the fixed cylinder, as well as corresponding free surface profiles and wave amplitudes, are discussed. The present method is found to be relatively straightforward, computationally effective and numerically stable for treating the problem of interactions among waves, viscous currents and bodies.

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

A time-dependent mild-slope equation for the extension of the classic mild-slope equation of Berkhoff is developed for the interactions of large ambient currents and waves propagating over an uneven bottom, using a Hamiltonian formulation for irrotational motions. The bottom topography consists of two components the slowly varying component which satisfies the mild-slope approximation, and the fast varying component with wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter describing the mild-slope condition. The theory is more widely applicable and contains as special cases the following famous mild-slope type equations: the classical mild-Slope equation, Kirby's extended mild-slope equation with current, and Dingemans's mild-slope equation for rippled bed. Finally, good agreement between the classic experimental data concerning Bragg reflection and the present numerical results is observed.

Propagation and interaction of ion-acoustic solitary waves in a quantum electron-positron-ion plasma

This paper discusses the existence of ion-acoustic solitary waves and their interaction in a dense quantum electron positron-ion plasma by using the quantum hydrodynamic equations. The extended Poincar^-Lighthill-Kuo perturbation method is used to derive the Korteweg-de Vries equations for quantum ion-acoustic solitary waves in this plasma. The effects of the ratio of positrons to ions unperturbation number density p and the quantum diffraction parameter He （Hp） on the newly formed wave during interaction, and the phase shift of the colliding solitary waves are studied. It is found that the interaction between two solitary waves fits linear superposition principle and these plasma parameters have significantly influence on the newly formed wave and phase shift of the colliding solitary waves. The investigations should be useful for understanding the propagation and interaction of ion-acoustic solitary waves in dense astrophysical plasmas （such as white dwarfs） as well as in intense laser-solid matter interaction experiments.

Solitary wave for a nonintegrable discrete nonlinear Schr?dinger equation in nonlinear optical waveguide arrays

We study a nonintegrable discrete nonlinear SchriSdinger （dNLS） equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term.

The Solitary Waves Solutions of the Internal Wave Benjamin-Ono Equation

The Benjamin-ono (BO) equation is an important nonlinear wave model which can describe the deep oceanic internal wave propagation. In this paper, the multi-algebraic solitary wave solutions for the internal wave BO equation including the linear velocity term in matrix form are given by the bilinear form. Based on the analytic solutions of the BO equation obtained in this paper and considering the hydrological parameters, the propagation of one-solitary wave and different kinds of interaction for the two-solitary waves are discussed and illustrated.

Consistent Riccati expansion solvability,symmetries,and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves

We study a forced variable-coefficient extended Korteweg-de Vries(KdV)equation in fluid dynamics with respect to internal solitary wave.Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevéexpansion.When the variable coefficients are time-periodic,the wave function evolves periodically over time.Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations.One-parameter group transformations and one-parameter subgroup invariant solutions are presented.Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method.The consistent Riccati expansion(CRE)solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE.Interaction phenomenon between cnoidal waves and solitary waves can be observed.Besides,the interaction waveform changes with the parameters.When the variable parameters are functions of time,the interaction waveform will be not regular and smooth.

Rogue waves of a(3+1)-dimensional BKP equation

We investigate certain rogue waves of a(3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method.We obtain semi-rational solutions in the determinant form,which contain two special interactions:(i)one lump develops from a kink soliton and then fuses into the other kink one;(ii)a line rogue wave arises from the segment between two kink solitons and then disappears quickly.We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time,which performs like a rogue wave.Furthermore,the higher-order semi-rational solutions describing the interaction between two lumps(one line rogue wave)and three kink solitons are presented.