Dispersive Wave System Research Articles

Construction of New Variable Separation Excitations via Extended Projective Ricatti Equation Expansion Method in (2+ 1)- Dimensional Dispersive Long Wave Systems

Utilizing the extended projective Ricatti equation expansion method, abundant variable separation solutions of the (2+1)-dimensional dispersive long wave systems are obtained. From the special variable separation solution (38) and by selecting appropriate functions, new types of interaction between the multi-valued and the single-valued solitons, such as semi-foldon and dromion, semi-foldon and peakon, semi-foldon and compacton are found. Meanwhile, we conclude that the solution v is essentially equivalent to the ’universal” formula (1).

LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS

We study nonlinear dispersive wave systems described by hyperbolic PDE's in ℝdand difference equations on the lattice ℤd. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

Oblique interactions of weakly nonlinear long waves in dispersive systems

Studies on the oblique interactions of weakly nonlinear long waves in dispersive systems are surveyed. We focus mainly our concentration on the two-dimensional interaction between solitary waves. Two-dimensional Benjamin–Ono (2DBO) equation, modified Kadomtsev–Petviashvili (MKP) equation and extended Kadomtsev–Petviashvili (EKP) equation as well as the Kadomtsev–Petviashvili (KP) equation are treated. It turns out that a large-amplitude wave can be generated due to the oblique interaction of two identical solitary waves in the 2DBO and the MKP equations as well as in the KP-II equation. Recent studies on exact solutions of the KP equation are also surveyed briefly.

Nonpropagating Solitary Waves in (2+1)-Dimensional Generalized Dispersive Long Wave Systems

An extended mapping approach is used to obtain a new type of variable separation excitation, with three arbitrary functions, of the 2+1-dimensional generalized dispersive long wave equation (DLWE). By selecting appropriate functions, the richness of nonpropagating solitons, such as nonpropagating dromion, nonpropagating ring, nonpropagating lump, and nonpropagating foldon, etc., is displayed for the ($2+1$)-dimensional generalized dispersive long wave equation (DLWE) in this paper. Meanwhile, we conclude that the solution v1 and v2 are essentially equivalent to the “universal” formula.

The Quasi-Nondispersive Regimes of Long Extratropical Baroclinic Rossby Waves over (Slowly Varying) Topography

Abstract Actual energy paths of long, extratropical baroclinic Rossby waves in the ocean are difficult to describe simply because they depend on the meridional-wavenumber-to-zonal-wavenumber ratio τ, a quantity that is difficult to estimate both observationally and theoretically. This paper shows, however, that this dependence is actually weak over any interval in which the zonal phase speed varies approximately linearly with τ, in which case the propagation becomes quasi-nondispersive (QND) and describable at leading order in terms of environmental conditions (i.e., topography and stratification) alone. As an example, the purely topographic case is shown to possess three main kinds of QND ray paths. The first is a topographic regime in which the rays follow approximately the contours f /hαc = a constant (αc is a near constant fixed by the strength of the stratification, f is the Coriolis parameter, and h is the ocean depth). The second and third are, respectively, “fast” and “slow” westward regimes little affected by topography and associated with the first and second bottom-pressure-compensated normal modes studied in previous work by Tailleux and McWilliams. Idealized examples show that actual rays can often be reproduced with reasonable accuracy by replacing the actual dispersion relation by its QND approximation. The topographic regime provides an upper bound (in general a large overestimate) of the maximum latitudinal excursions of actual rays. The method presented in this paper is interesting for enabling an optimal classification of purely azimuthally dispersive wave systems into simpler idealized QND wave regimes, which helps to rationalize previous empirical findings that the ray paths of long Rossby waves in the presence of mean flow and topography often seem to be independent of the wavenumber orientation. Two important side results are to establish that the baroclinic string function regime of Tyler and Käse is only valid over a tiny range of the topographic parameter and that long baroclinic Rossby waves propagating over topography do not obey any two-dimensional potential vorticity conservation principle. Given the importance of the latter principle in geophysical fluid dynamics, the lack of it in this case makes the concept of the QND regimes all the more important, for they are probably the only alternative to provide a simple and economical description of general purely azimuthally dispersive wave systems.

New variable separation excitations of (2+1)-dimensional dispersive long-water wave system obtained by an extended mapping approach☆

By means of an extended mapping approach, a new type of variable separation excitation with three arbitrary functions of the (2 + 1)-dimensional dispersive long-water wave system (DLW) is derived. Based on the derived variable separation excitation, abundant localized structures such as dromion, ring, peakon and foldon etc. are re-revealed by selecting appropriate functions in this paper.

Nonpropagating Solitons in (2+1)-Dimensional Dispersive Long-Water Wave System

With the help of an extended mapping approach, a new type of variable separation excitation with three arbitrary functions of the(2+1)-dimensional dispersive long-water wave system (DLW) is derived. Based on the derived variable separation excitation, abundant non-propagatingsolitons such as dromion, ring, peakon, and compacton etc. are revealed by selecting appropriate functions in this paper.

New variable separation excitations of (2+1)-dimensional dispersive long-water wave system obtained by an extended mapping approach

Abstract By means of an extended mapping approach, a new type of variable separation excitation with three arbitrary functions of the (2 + 1)-dimensional dispersive long-water wave system (DLW) is derived. Based on the derived variable separation excitation, abundant localized structures such as dromion, ring, peakon and foldon etc. are re-revealed by selecting appropriate functions in this paper.

Multi-valued solitary waves in multidimensional soliton systems

Considering that folded phenomena are rather universal in nature and some arbitrary functions can be included in the exact excitations of many (2+1)-dimensional soliton systems, we use adequate multivalued functions to construct folded solitary structures in multi-dimensions. Based on some interesting variable separation results in the literature, a common formula with arbitrary functions has been derived for suitable physical quantities of some significant (2+1)-dimensional soliton systems like the generalized Ablowitz?Kaup?Newell?Segur (GAKNS) model, the generalized Nizhnik?Novikov?Veselov (GNNV) system and the new (2+1)-dimensional long dispersive wave (NLDW) system. Then a new special type of two-dimensional solitary wave structure, i.e. the folded solitary wave and foldon, is obtained. The novel structure exhibits interesting features not found in the single valued solitary excitations.

The Effect of Meridional and Vertical Shear on the Interaction of Equatorial Baroclinic and Barotropic Rossby Waves

Simplified asymptotic equations describing the resonant nonlinear interaction of equatorial Rossby waves with barotropic Rossby waves with significant midlatitude projection in the presence of arbitrary vertically and meridionally sheared zonal mean winds are developed. The three mode equations presented here are an extension of the two mode equations derived by Majda and Biello [1] and arise in the physically relevant regime produced by seasonal heating when the vertical (baroclinic) mean shear has both symmetric and antisymmetric components; the dynamics of the equatorial baroclinic and both symmetric and antisymmetric barotropic waves is developed. The equations described here are novel in several respects and involve a linear dispersive wave system coupled through quadratic nonlinearities. Numerical simulations are used to explore the effect of antisymmetric baroclinic shear on the exchange of energy between equatorial baroclinic and barotropic waves; the main effect of moderate antisymmetric winds is to shift the barotropic waves meridionally. A purely meridionally antisymmetric mean shear yields highly asymmetric waves which often propagate across the equator. The two mode equations appropriate to Ref. [1] are shown to have analytic solitary wave solutions and some representative examples with their velocity fields are presented.

Determination of boundaries of unsteady oscillatory zone in asymptotic solutions of non-integrable dispersive wave equations

We propose a simple general method for analytic determination of the boundaries of the expanding nonlinear oscillation zone occurring in the decay of a step problem for non-integrable dispersive wave equations. A remarkable feature of the method is that it essentially uses only the dispersionless limit and the linear dispersion relation of the original nonlinear dispersive wave system. A concrete example pertaining to collisionless plasma dynamics is considered and complete agreement with the results of earlier numerical simulations is demonstrated.

Chaotic Dynamical Behaviour in Soliton Solutions for a New(2+1)-Dimensional Long Dispersive Wave System

With the help of variable separation approach, a quite generalexcitation of a new (2+1)-dimensional long dispersive wave systemis derived. The chaotic behaviour, such as chaotic line solitonpatterns, chaotic dromion patterns, chaotic-period patterns, andchaotic-chaotic patterns, in some new localized excitations are foundby selecting appropriate functions.

Fractal Dromion, Fractal Lump, and Multiple Peakon Excitations in a New (2 + 1)-Dimensional Long Dispersive Wave System**The project supported by the Foundation of “151 Talent Engineering” of Zhejiang Province and the Natural Science Foundation of Zhejiang Province of China under Grant No. 100039

By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, fractal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.

Effectiveness of the continuous wavelet transform in the analysis of some dispersive elastic waves

Although there have been many investigations employing the continuous wavelet transform for the analysis of dispersive waves, they seem to lack theoretical justifications for the effectiveness of the continuous wavelet transform (CWT) over other time–frequency analysis tools such as the short-time Fourier transform (STFT). The goal in this paper is to offer theoretical and experimental justifications for its effectiveness by comparing the performance of CWT and STFT in terms of their time–frequency analysis capabilities of certain dispersive elastic waves. The waves in consideration are elastic flexural waves generated by an impact in a solid circular cylinder. The ridge analysis procedure is employed to estimate instantaneous frequencies by CWT and STFT. Although in the present investigation we are focused on a limited class of dispersive waves, it gives an insight into the effectiveness of CWT for the analysis of other types of dispersive wave systems.

Proliferation scheme for Kaup–Boussinesq system

The proliferation scheme for Kaup–Boussinesq system (dispersive water wave system) is discussed. The recurrent procedure gives the series of integrable nonlinear equations coupled by Miura maps and its Lax pairs. New integrable nonlinear systems are found in this series.

Resonances in Dispersive Wave Systems

Weakly nonlinear wave interactions under the assumption of a continuous, as opposed to discrete, spectrum of modes is studied. In particular, a general class of one‐dimensional (1‐D) dispersive systems containing weak quadratic nonlinearity is investigated. It is known that such systems can possess three‐wave resonances, provided certain conditions on the wavenumber and frequency of the constituent modes are met. In the case of a continuous spectrum, it has been shown that an additional condition on the group velocities is required for a resonance to occur. Nonetheless, such so‐called double resonances occur in a variety of physical regimes. A direct multiple scale analysis of a general model system is conducted. This leads to a system of three‐wave equations analogous to those for the discrete case. Key distinctions include an asymmetry between the temporal evolution of the modes and a longer time scale of as opposed to O(εt). Extensions to additional dimensions and higher‐order nonlinearities are then made. Numerical simulations are conducted for a variety of dispersions and nonlinearities providing qualitative and quantitative agreement.

On the stability of solitary waves with decaying oscillatory tails

In dispersive wave systems with dispersion relations such that the phase speed attains an extremum at a finite wavenumber, a rich variety of solitary waves that feature decaying oscillatory tails i...

Do envelope solitons radiate

In dispersive wave systems, when leading-order nonlinear and dispersive effects, are taken into account the envelope of a small-amplitude narrow-band wave pulse is known to satisfy the nonlinear Schrodinger (NLS) equation which, under certain conditions, admits envelope-soliton solutions. These solitons describe locally confined wave groups with envelopes of permanent form and find applications in various physical contexts. Here, is addressed the question of whether NLS envelope solitons survive when higher-order effects are taken into account. Based on a kinematic argument first, it is suggested that oscillatory tails are inevitably emitted, and this claim is further supported by numerical computations by use of a fifth-order Korteweg-deVries equation as a simple example. The radiation of tails is caused by a resonance mechanism that lies beyond all orders of the usual multiple-scale expansion leading to the NLS equation, and a procedure for calculating these tails by use of exponential asymptotics is outlined. Despite having exponentially small amplitude in the asymptotic sense, the radiated tails can be significant when pulses of relatively short duration are considered.

A Note on Supersymmetric Two-Boson Equation11This project supported by National Natural Science Foundation of China

We show that the supersymmetric version of two-boson or dispersive water wave system is nothing but one of the systems. We also consider a constrained , which is a gauge equivalence of it.

Some Properties of Nonlinear Wave Systems

The results of a preliminary study of a particular nonlinear system of partial differential equations are presented. While much of this work pertains to two coupled nonlinear Schrödinger equations, it is believed that the properties found are representative of many other dispersive wave systems.