Generation and evolution of oblique solitary waves in supercritical flows

It is considered that a thin strut sits in a supercritical shallow water flow sheet over a homogeneous or very mildly varying topography. This stationary 3-D problem can be reduced from a Boussinesq-type equation into a KdV equation with a forcing term due to uneven topography, in which the transverse coordinate Y plays a same role as the time in original KdV equation. As the first example a multi-soliton wave pattern is shown by means of N-soliton solution. The second example deals with the generation of solitary wave-train by a wedge-shaped strut on an even bottom. Whitham's average method is applied to show that the shock wave jump at the wedge vertex develops to a cnoidal wave train and eventually to a solitary wavetrain. The third example is the evolution of a single oblique soliton over a periodically varying topography. The adiabatic perturbation result due to Karpman & Maslov (1978) is applied. Two coupled ordinary differential equations with periodic disturbance are obtained for the soliton amplitude and phase. Numerical solutions of these equations show chaotic patterns of this perturbed soliton.     

Modeling the interaction of solitary waves in magnetic tubes

The Cauchy problems of the propagation of a single wave and the interaction of two solitary waves of different amplitude are solved numerically for the case of slow symmetric surface waves in a magnetic tube. It is found that the solitary waves interact in the same way as the solitons of the known soliton equations such as the Korteweg-de Vries and Benjamin-Ono equations, i.e., preserve their shape after interacting. The way in which the solitons decrease at infinity is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 183–186, March–April, 1989.The author wishes to thank M. S. Ruderman for formulating the problem and V. B. Baranov for his interest in the work.     

General solution for interaction of solitary waves including head-on collisions

A corrected version of the Boussinesq equation for long water waves is derived and its general solution for interaction of any number of solitary waves, including head-on collisions, is given. For two solitary waves in head-on collision (which includes the case of normal reflection) the results agree with the experiments known.     

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 solution 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 solution (1973) of the Boussinesq equation, to which the present solution is compared for the simplest case of two solitary waves in head-on collision.     

Numerical simulation of interaction of solitary deformation waves in microstructured solids

In the present paper 1D wave propagation in microstructured solids is modelled based on the Mindlin theory and hierarchical approach. The governing equation under consideration is non-integrable therefore it is analysed numerically. Propagation and interaction of localised initial pulses is simulated numerically over long time intervals by employing the pseudospectral method. Special attention is paid to the solitonic character of the solution.     

The stability of finite-amplitude interfacial solitary waves

The linear stability of finite-amplitude interfacial gravity solitary waves propagating in a two-layer fluid is investigated analytically focusing on the occurrence of an exchange of stability. We make an asymptotic analysis for small growth rates of infinitesimal disturbances, and explicitly obtain their growth rates near an exchange of stability. The result indicates that an exchange of stability occurs at every stationary value of the total energy of the solitary waves. It also gives us information whether the number of growing modes increases or decreases after experiencing the exchange of stability. We apply these analytical results to specific interfacial solitary waves, and find various features on their stability that are not seen in the case of surface solitary waves.     

Mode interaction in rotating Rayleigh–Bénard convection

The results of direct numerical simulations of convection in a uniformly rotating vertical cylinder with no-slip boundary conditions are described, and used to study the dynamics associated with transitions between states with adjacent azimuthal wave numbers far from onset. In certain regimes a novel burst-like state is identified and described. A dynamical-systems explanation for this behavior is suggested.     

Modeling the interaction of solitary waves and semi-circular breakwaters by using unsteady reynolds equations

A vertical 2 -D numerical wave model was developed based on unsteady Reynolds equations. In this model, the k-epsilon models were used to close the Reynolds equations, and volume of fluid( VOF) method was used to reconstruct the free surface. The model was verified by experimental data. Then the model was used to simulate solitary wave interaction with submerged, alternative submerged and emerged semi-circular breakwaters. The process of velocity field, pressure field and the wave surface near the breakwaters was obtained. It is found that when the semi-circular breakwater is submerged, a large vortex will be generated at the bottom of the lee side wall of the breakwater ; when the still water depth is equal to the radius of the semi-circular breakwater, a pair of large vortices will be generated near the shoreward wall of the semi-circular breakwater due to wave impacting, but the velocity near the bottom of the lee side wall of the breakwater is always relatively small. When the semi-circular breakwater is emerged, and solitary wave cannot overtop it, the solitary wave surface will run up and down secondarily during reflecting from the breakwater. It can be further used to estate the diffusing and transportation of the contamination and transportation of suspended sediment.     

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow. The Project Supported by the National Natural Science Foundation of China.     

Stability of solitary waves in dispersive media described by a fifth-order evolution equation

The problem of the existence and dynamical stability of solitary wave solutions to a fifth-order evolution equation, generalizing the well-known Korteweg-de Vries equation, is treated. The theoretical framework of the paper is largely based on a recently developed version of positive operator theory in Fréchet spaces (which is used for the existence proof) and the theory of orbital stability for Hamiltonian systems with translationally invariant Hamiltonians. The validity of sufficient conditions for stability are established. The shape of solitary waves under analysis are determined by a numerical solution of the boundary-value problem followed by a correction using the Picard method of 4–12 orders of accuracy.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

On solitary waves in plasma

From the equations of hydrodynamics and electrodynamics, a system of coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exists for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The interaction of the longitudinal solitary waves with each other is studied and an upper bound on the amplitudes of these waves is obtained. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.     

The strain solitary waves in a nonlinear elastic rod

Solitary strain waves in a nonlinear elastic rod are analysed in this paper; influence of the physical and geometrical parameters of the rod on the waves are discussed; some main properties of the solitary waves are pointed out.     

Cubic non-linearity and longitudinal surface solitary waves

It is shown that for some seismic media both quadratic and cubic non-linearities should be taken into account in the governing equation for longitudinal waves. The new equation is obtained to account for non-linear surface waves in a medium surrounding a non-linearly elastic rod. Exact solutions of the equation allow us to describe simultaneous propagation of tensile and compressive localized strain waves. Various interactions between these waves give rise to both the multi-bump and “Mexican hat” localized wave structures closer to the surface waves recently observed in experiments.     

Correlation between theoretical and experimental solitary waves

Experimental data on surface solitary waves generated by five methods are given. These data and literature information show that at amplitudes 0.2<a/h<0.6 (h is the initial depth of the liquid), experimental solitary waves are in good agreement with their theoretical analogs obtained using the complete model of liquid potential flow. Some discrepancy is observed in the range of small amplitudes. The reasons why free solitary waves of theoretically limiting amplitude have not been realized in experiments are discussed, and an example of a forced wave of nearly limiting amplitude is given. The previously established fact that during evolution from the state of rest, undular waves break when the propagation speed of their leading front reaches the limiting speed of propagation of a solitary wave is confirmed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 44–52, May–June, 1999.     

The splitting of solitary waves running over a shallower water

The Korteweg-de Vries type of equation (called KdV-top) for uni-directional waves over a slowly varying bottom that has been derived by Van Groesen and Pudjaprasetya [E. van Groesen, S.R. Pudjaprasetya, Uni-directional waves over slowly varying bottom. Part I. Derivation of a KdV-type of equation, Wave Motion 18 (1993) 345–370.] is used to describe the splitting of solitary waves, running over shallower water, into two (or more) waves. Results of numerical computations with KdV-top are presented; qualitative and quantitative comparisons between the analytical and numerical results show a good agreement.     

The propagation of solitary waves in a nonlinear elastic rod

In this paper, the inverse scattering method is used to analyse strain sohtarv waves bed nonlinear clastic rod. Properties of solitary waves and their influence on solid structures are discussed in detail. Some quantitative results are given.     

The numerical study of roll‐waves in inclined open channels and solitary wave run‐up

In this paper, we introduce a finite‐volume kinetic BGK scheme and its applications to the study of roll and solitary waves. The current scheme is based on the numerical solution of the gas‐kinetic Bhatnagar–Gross–Krook model in the flux evaluation across each cell interface. An intrinsic connection between the BGK model and time‐dependent, non‐linear, non‐homogeneous shallow‐water equations enables us to solve shallow‐water equations automatically with our kinetic scheme. The analytical solution, experimental measurements, and numerical calculations for problems associated with roll‐waves down an inclined open channel and solitary waves incident on a sloped beach are also presented. Copyright © 2005 John Wiley & Sons, Ltd.