Bilinear form and soliton interactions for the modified Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics

In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transformation and auxiliary function, new bilinear form for the mKP equation is constructed, which is different from those in previous literatures. Based on the bilinear form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.     

Bilinear Bäcklund transformation,Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Bäcklund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Bäcklund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

     

Solitary waves in media with dispersion and dissipation (a review)

The latest results relating to the theory of nonlinear waves in dispersive and dissipative media are reviewed. Attention is concentrated on small-amplitude solitary waves and, in particular, on the classification of types of solitary waves, their conditions of existence, the evolution of local perturbations associated with the presence of solitary waves of various types, and problems of the existence of nonlinear waves localized with respect to a particular direction as the space dimension increases (spontaneous dimension breaking). As examples of dispersive and dissipative media admitting plane solitary waves of various types, we consider a cold collisionless plasma, an ideal incompressible fluid of finite depth beneath an elastic plate and with surface tension, and a fluid in a rapidly oscillating rectangular vessel (Faraday resonance). Examples of spontaneous dimension breaking are considered for the generalized Kadomtsev-Petviashvili equation. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–27. March–April, 2000. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 99-0101150).     

Symmetry and Uniqueness of the Solitary-Wave Solution for the Ostrovsky Equation

We consider herein the Ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water, or the capillary waves in a plasma with the effects of rotation. Using the modified sliding method, we prove that the solitary wave moving to the left to the Ostrovsky equation is symmetric about the origin and unique up to translations. We also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed, weak rotation, and dispersive parameter.     

Solitary wave trains in a cold plasma

The existence of traveling solitary waves, the products of modulation instability in a cold quasi-neutral plasma, is considered. Solitary waves of this type (solitary wave trains) are formed as a result of bifurcation from a nonzero wave number of the linear wave spectrum. It is shown that the complete system of equations describing the wave process in a cold plasma has solutions of the solitary wave train type, at least when the undisturbed magnetic field is perpendicular to the wave front. Sufficient conditions of existence of solitary wave trains in weakly dispersive media are also formulated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 154–161, September–October, 1996.     

INTRACELLULAR SOLITARY PULSE CALCIUM WAVES IN FROG SYMPATHETIC NEURONS

IntroductionCalciumionlevelsinintracellularsympatheticneuronsplayacontrollingroleinthetransmissionprocessesofvarioussignals,includingneuron_transmittersecretion ,regulationofmembraneexcitability ,inductionofgeneexpression ,etc .[1]However,howisthecytoplasmiccalciumconcentrationinneuronsregulated ?WehavealreadyknownthattheraiseincalciumconcentrationinXenopuslaevisoocyteismainlyoriginatedfromtheinfluxofCa2 outofcalciumstoresintheendoplasmicreticulum ,whichisinitiatedwiththestimulationofextrace…     

Evolution of ion–ion acoustic instability in multi-ion plasma sheaths

Nonlinear Dynamics - The generation of ion-acoustic solitary waves is investigated in a nonuniform multicomponent collisional plasma sheath containing cold ions and Boltzmann electrons to probe the...     

Weakly nonlocal envelope solitary waves: numerical calculations for the Klein-Gordon (φ) equation

“Weakly nonlocal” solitary waves differ from ordinary solitary waves by possessing small amplitude, oscillatory “wings” that extend indefinitely from the large amplitude “core”. Such generalized solitary waves have been discovered in capillarygravity water waves, particle physics models, and geophysical Rossby waves. In this work, we present explicit calculations of weakly nonlocal envelope solitary waves. Each is a sine wave modulated by a slowly-varying “envelope” that itself propagates at the group velocity. Our example is the cubically nonlinear Klein-Gordon equation, which is a model in particle physics (φ⁴ theory) and in electrical engineering (with a different sign). Both cases have weakly nonlocal“breather” solitons. Via the Lorentz invariance, each breather generates a one-parameter family of nonlocal envelope solitary waves. The φ⁴ breather was described and calculated in earlier work. This generates envelope solitons which have “wings” that are (mostly) proportional to the second harmonic of the sinusoidal factor. In this article, we calculate breathers and envelope solitary waves for the second, electrical engineering case. Since these, unlike the φ⁴ waves, contain only odd harmonics, the envelope solitary waves are nonlocal only via the third harmonic.     

Formation of solitary waves on gas-sheared liquid layers

The origin of solitary waves on gas-liquid sheared layers is studied by comparing the behavior of the wave field at sufficiently low liquid Reynolds number, R_L, where solitary waves are observed to form, to measurements at higher R_L where solitary waves do not occur. Observations of the wave field with high-speed video imaging suggest that solitary waves, which appear as a secondary transition of the stratified gas-liquid interface, emanate from existing dominant waves, but that not all dominant waves are transformed. From measurements of interface tracings it is found that for low R_L, waves which have amplitude/substrate depth (a/h) ratios of 0.5–1 occur while for higher R_L, no such waves are observed. A comparison of amplitude/wavelength ratios shows no distinction for different R_L. Consequently, it is conjectured that solitary waves originate from waves with sufficiently large a/h ratios; this change of form being similar to wave breaking. The dimensionless wavenumber is found to be smaller at low R_L, where solitary waves are observed. This suggests that perhaps, larger precursor (to solitary wave) waves are possible because the degree of dispersion, which acts to break waves into separate modes, is lower.     

Solitary and extended waves in the generalized sinh-Gordon equation with a variable coefficient

New solitary and extended wave solutions of the generalized sinh-Gordon (SHG) equation with a variable coefficient are found by utilizing the self-similar transformation between this equation and the standard SHG equation. Two arbitrary self-similar functions are included in the known solutions of the standard SHG equation, to obtain exact solutions of the generalized SHG equation with a specific variable coefficient. Our results demonstrate that the solitary and extended waves of the variable-coefficient SHG equation can be manipulated and controlled by a proper selection of the two arbitrary self-similar functions.     

Derivation of the Zakharov Equations

This article continues the study, initiated in [27, 7], of the validity of the Zakharov model which describes Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for prepared initial data. We apply this result to the Euler–Maxwell equations which describes laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler–Maxwell equations in terms of WKB approximate solutions, the leading terms of which are solutions of the Zakharov equations. Due to the transparency properties of the Euler–Maxwell equations evidenced in [27], this study is carried out in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of JOLY, MéTIVIER and RAUCH [12, 13]; recent results by LANNES on norms of pseudodifferential operators [14]; and a semiclassical paradifferential calculus.     

Problem of Breakdown of Solitary Waves and Discontinuities

The evolution of initial data of the solitary-wave type with time is investigated numerically. The solitary wave amplitude decreases due to the generation of short-wave radiation. This solution is interpreted as the solution with a discontinuity qualitatively analogous to the solution of the problem of the breakdown of an arbitrary discontinuity in dissipationless systems. The solitary wave amplitude reduction rate is estimated, first for a generalized Korteweg-de Vries equation and then for plasma waves. Features of the investigation are analyzed for cold and hot-electron plasmas.     

Weak oblique collisions of interfacial solitary waves

A third order perturbation solution is developed to describe the interaction between two solitary waves approaching each other at an angle close to 180 ° on the interface between two immiscible inviscid homogeneous fluids. The solution is steady in the frame of reference moving with the point of intersection of the waves. To lowest order, the solution consists simply of the superposition of the undisturbed solitary waves. Second-order collision effects include interaction terms localized near the point of intersection and a phase shift in the solitary waves. In addition to corrections to the phase shift and localized interaction terms, third order effects are found to include a wave train that is stationary in the frame of reference moving with the point of intersection of the solitary waves. The amplitudes of the wave train and localized interaction terms diminish with distance from the point of intersection, and the solitary waves recover their initial shape asymptotically long after the collision. Thus, the only long-term effect of the collision is a phase shift.     

Criticality manifolds and their role in the generation of solitary waves for two-layer flow with a free surface

The role of criticality manifolds is explored both for the classification of all uniform flows and for the bifurcation of solitary waves, in the context of two fluid layers of differing density with an upper free surface. While the weakly nonlinear bifurcation of solitary waves in this context is well known, it is shown herein that the critical nonlinear behaviour of the bifurcating solitary waves and generalized solitary waves is determined by the geometry of the criticality manifolds. By parametrizing all uniform flows, new physical results are obtained on the implication of a velocity difference between the two layers on the bifurcating solitary waves.     

On the backward Lamb waves near thickness resonances in anisotropic plates

A closed-form expression for the leading-order dispersion coefficient, describing the trend of Lamb-wave branches at their onset from thickness resonances, is derived for an arbitrary anisotropic plate. The sign of this coefficient and hence of the in-plane group velocity near cutoffs decides the existence or non-existence of the backward Lamb waves without a necessity to calculate the dispersion branches. A link between the near-cutoff dispersion of Lamb waves and the curvature of bulk-wave slowness curves in a sagittal plane is analyzed. It is established that a locally concave slowness curve of a bulk mode entails the backward Lamb waves at the onset of branches emerging from this bulk mode resonances of high enough order. A simple sufficient condition for no backward Lamb waves near the resonances associated with a convex slowness curve is also noted. Two special cases are discussed: the first involves the coupled resonances of degenerate bulk waves, and the second concerns quasi-degenerate resonances which give rise to pairs of dispersion branches with a quasilinear positive and negative onset. Occasions of the backward Lamb waves in isotropic plate materials are tabulated.     

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.     

Nonlinear dynamical magnetosonic wave interactions and collisions in magnetized plasma

The one-dimensional nonlinear dynamical wave interactions in a system of quasineutral two-fluid plasma in a constant magnetic field are investigated.The existence of the travelling wave solutions is discussed.The modulation stability of linear waves and the modulation instability of weakly nonlinear waves are presented.Both suggest that the Korteweg-de Vries(KdV) system is modulationally stable.Besides,the wave interactions including the periodic wave interaction and the solitary wave interaction are captured and presented.It is shown that these interacting waves alternately exchange their energy during propagation.The Fourier spectrum analysis is used to depict the energy transformation between the primary and harmonic waves.It is known that the wave interactions in magnetized plasma play an important role in various processes of heating and energy transportation in space and astrophysical plasma.However,few researchers have considered such magnetohydrodynamic(MHD) wave interactions in plasma.It is expected that this work can provide additional insight into understanding of behaviors of MHD wave interactions.     

Dynamical behavior of traveling wave solutions of ion acoustic plasma equations

By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.     

Experimental Study of Strongly Nonlinear Resonances and Anti-Resonances in Granular Dimer Chains

Previous theoretical works considered the intrinsic dynamics of one-dimensional uncompressed granular dimer (diatomic) chains composed of pairs of dissimilar spherical elastic beads in Hertzian interaction. Such ordered granular media exhibit essentially nonlinear acoustics and have been characterized as ‘sonic vacua’ due to the fact that the speed of sound in these media (as defined in classical acoustics) is zero. Yet, depending on the mass ratios of the pairs of dissimilar beads of these dimers, it was proven that they may possess countable infinities of anti-resonances leading to solitary waves (this in spite of their high inhomogeneity), or countable infinities of strongly nonlinear resonances leading to passive strong attenuation of propagating pulses through energy radiation by means of excitation of traveling waves. The aim of this work is to experimentally verify the existence of these strongly nonlinear dynamics through a series of experiments involving granular dimer chains supported by flexures. By carefully designing the supporting flexures so that their dynamics is sufficiently ‘soft’ and thus separate from the ‘stiff’ dynamics governing the bead to bead interactions, we overcome a basic limitation for the experimental realization of such dimer systems, namely the construction of one-dimensional dimer chains with beads of different radii. Our results confirm experimentally the occurrence of nonlinear resonances and anti-resonances in dimer chains, and conclusively prove the capacity of appropriately designed granular dimers for passive strong attenuation of propagating pulses due to nonlinear resonance. Moreover, we validate the theoretical prediction that within the elastic range of bead to bead dynamical interactions the results are fully re-scalable with respect to energy. This work provides the first experimental evidence of strongly nonlinear resonances and anti-resonances in essentially nonlinear ordered granular media.