Envelope Soliton Solutions Research Articles

Envelope solitons with stationary crests

Recent analytical and numerical work has shown that gravity–capillary surface waves as well as other dispersive wave systems support symmetric solitary waves with decaying oscillatory tails, which bifurcate from linear periodic waves at an extremum value of the phase speed. It is pointed out here that, for small amplitudes, these solitary waves can be interpreted as particular envelope-soliton solutions of the nonlinear Schrödinger equation, such that the wave crests are stationary in the reference frame of the wave envelope. Accordingly, these waves (and their three-dimensional extensions) are expected to be unstable to oblique perturbations.

Axisymmetric waves on a vertical vortex in a stratified fluid

Axisymmetric wave propagation along a vertical vortex core in a stably stratified fluid is considered theoretically. The fluid is assumed to be inviscid, incompressible, nondiffusive, and exponentially stratified. A linear analysis under the Boussinesq approximation shows that discrete inertial modes (bounded modes) are allowed in addition to continuous internal gravity waves (unbounded modes), when the stratification is not too strong. These inertial modes, whose eigenfunctions are confined to the vorticity region, disappear if the Brunt–Väisälä frequency N2 exceeds the maximum value of the Rayleigh function. Concrete results are given for the Burgers vortex. A weakly nonlinear analysis indicates that inertial modes (if permitted) are generated through the resonant interactions between two internal gravity waves. The time evolution of its amplitude is described by a cubic nonlinear Schrödinger equation, which admits envelope soliton solutions for shorter carrier waves only, viz., the soliton window has a low wave-number cutoff.

Envelope Soliton as an Intrinsic Localized Mode in a One-Dimensional Nonlinear Lattice

An envelope soliton in a nonlinear lattice has been studied theoretically and numerically. First, the nonlinear Schrodinger equation describing a lattice wave with large wave number is derived and then the equation of motion for the nonlinear lattice is solved numerically taking the one envelope soliton solution of the nonlinear Schrodinger equation as an initial wave. It is shown that the envelope soliton with zero group velocity exists stably in the lattice and that the frequency of carrier wave is above the cut-off frequency of the lattice. An intrinsic localized mode in the nonlinear lattice developed by Takeno is pointed out to be the envelope soliton with zero group velocity.

Dark envelope solitons of fast magnetosonic surface waves in solar flux tubes

We have studied the modulational stability of a finite-amplitude fast sausage magnetosonic surface wave traveling along a thin magnetic slab in the solar photosphere (chromosphere). The equation governing the evolution of the fast-wave envelope modulated by a slow wave driven by the ponderomotive force proves to be the cubic nonlinear Schrodinger equation which in photospheric conditions admits only dark envelope soliton solutions. The possibility of the existence of such solitary waves in the solar atmosphere is discussed.

Asymmetric Discrete Envelope Soliton Solutions for Semi-Discrete Nonlinear Wave Equations

One and four types of asymmetric discrete envelope soliton solutions are found for semi-discrete sine-Gordon equation and semi-discrete wave equation with cubic nonlinearity. They are found by appropriately transforming the variables of already-known discrete envelope soliton solutions. Some properties of these solutions are discussed.

Low-amplitude breather and envelope solitons in quasi-one-dimensional physical models.

In the low-amplitude limit we examine the breather- and envelope-soliton solutions of the generalized nonlinear Klein-Gordon system as nonlinear Schr\odinger solitons. The existence of breather and envelope solutions is determined in the continuum and quasidiscrete limit; in this case the oscillations of the carrier in the envelope are treated exactly. The results are applied to the perturbed sine-Gordon and ${\ensuremath{\varphi}}^{4}$ systems; in both cases the asymmetry of the breathers is controlled by the amplitude of the external force. The study is generalized to the calculations of low-amplitude breather modes in a ferromagnetic chain with a small out-of-plane angle.

Discrete Model for Two-Component Wave Equations with Cubic Nonlinearity

A method of discretization along the light-cone is applied to two-component nonlinear wave equations found by Sarker, Trullinger and Bishop and Montonen. We present a soliton solution, envelope soliton solutions and elliptic function solutions for discretized wave equations of coupled real fields, and complex envelope soliton solutions for coupled real and complex fields.

Properties of Alfvén Solitons in a Finite-Beta Plasma J. Plasma Phys. vol. 27, 1982, p. 193

In the aforementioned paper we obtained an equation for non-linear Alfvén waves in a finite-β plasma, and investigated envelope soliton solutions thereof. The purpose of this note is to point out an error in the derivation of the soliton envelopes, and present corrected expressions for these solitons.The error arises from our assumption of translational invariance of both the envelope and phase of an envelope soliton expressed in equations (16) and (17). Rather, the phase is related to the amplitude by where y ≡ x — VEt is a comoving co-ordinate, and all other quantities are defined in the above paper.

Evolution theorem for a class of perturbed envelope soliton solutions

Envelope soliton solutions of a class of generalized nonlinear Schrödinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift. For ∂η2N >0, the system is stable in the sense of Liapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relatons, and others.

Alfvén solitons in the solar wind

The interaction of circularly‐polarized Alfvén waves with the surrounding plasma in high speed solar wind streams is investigated. Alfvén wave modulational instability is discussed and nonlinear envelope soliton solutions of the magnetohydrodynamic equations are introduced. The characteristics of these Alfvén solitons are compared with observational results obtained from Helios I and II. A model of the expected turbulent spectrum due to a collection of such solitons is constructed and its radial dependence is investigated, again along with comparison to Helios data.

Scattering analysis and synthesis of wave trains

AbstractThe Zakharov-Shabat scattering transform is an exact solution technique for the nonlinear Schrödinger equation, which describes the time evolution of weakly nonlinear wave trains. Envelope soliton and uniform wave train solutions of the nonlinear Schrödinger equation are separable in scattering transform space. The scattering transform is a potential analysis and synthesis technique for natural wave trains. Discrete versions of the direct and inverse scattering transform are presented, together with proven algorithms for their numerical computation from typical ocean wave records. The consequences of discrete resolution are considered.

Perturbation of solitons in the classical continuum isotropic Heisenberg spin system

The dynamics of a one-dimensional classical continuum isotropic Heisenberg ferromagnetic spin system in the presence of a weak relativistic interaction, which causes damping of the spin motion, is considered. The corresponding evolution equation is identified with a damped nonlinear Schrödinger equation in terms of the energy and current densities of the unperturbed system. A direct perturbation method, along the lines of Kodama and Ablowitz, is developed for the envelope soliton solution of the nonlinear Schrödinger equation and the explicit perturbed solution obtained. This solution is found to be valid in a finite domain of the propagation space. To cover the entire region, a uniform solution is constructed using the matched asymptotic expansion technique. Finally, the spin vectors are constructed using the known procedures in differential geometry and the consequences of damping analysed briefly.

Nonlinear evolution of the transverse instability of plane-envelope solitons

The nonlinear evolution of the transverse instability of plane envelope soliton solutions of the nonlinear Schrödinger equation is investigated. For the case where the spatial derivatives in the two-dimensional nonlinear Schrödinger equation are elliptic a critical transverse wavenumber is found above which the soliton is stable. Application of the multiple time scale method near this critical transverse wavenumber gives a dynamical equation for the nonlinear evolution of the transverse instability. Nonlinearity is found to enhance the growth of the linearly unstable mode. The results are discussed in connection with soliton collapse.

Natural Wave Trains and Scattering Transform

The potential of the nonlinear Schroedinger equation and the Zakharov-Shabat scattering transformation as a third generation analysis technique for natural ocean wave trains is considered. Such an approach retains many of the advantages of the second generation Gaussian random wave model but accomodates such fundamentally nonlinear concepts as Benjamin-Feir instability and especially envelope solitons. Wave envelope solitons are naturally occurring and stable wave groups which exist quite separately from uniform wave train solutions. Envelope soliton and uniform wave train solutions interact nonlinearly in physical space and are difficult to separately identify from a wave record. Resolution is possible, however, in scattering transform space. The direct scattering transform leads to the scattering data, a broadly analogous summary of a wave record to the Fourier spectrum. The inverse scattering transform recreates the wave train at the same or a later time, in a similar manner to the inverse Fourier transform. The results of exploratory computations of both analysis and synthesis of wave trains are included, together with a discussion of the potential application of the method to natural wave trains.

Exact envelope-soliton solutions of a two-dimensional nonlinear wave equation

ExactN-envelope-soliton solutions are obtained, by extending Hirota's procedure, for the twodimensional nonlinear wave Eqn. (1) withq>0, which describes the evolution of the envelope of a train of surface gravity waves on deep water. They are shown to propagate in directions making an angle greater than tan−1/√2 with the propagation direction of the underlying carrier waves. We also point out and discuss the limitations of Hirota's procedure for generating solitonsolutions to problems of more than one spatial dimensions. Envelope-soliton solutions to Eqn. (1) withq<0 are also discussed.

Reduction of the Boussinesq Type of Equation to Modified Hirota Equation

A model equation as above (Boussinesq form) which governs the lattice wave and shallow water waves, has been reduced to a modified form of Hirota equation for intermediate range of wave number ( k <1), considering the case when the central wave number k and the spectral width Δ k ∼ e of the wave packet are both smaller than unity but of comparable magnitude. For the modified Hirota equation, the envelope soliton solution is obtained and the condition for modulational instability is derived. This is found to differ appreciably from the corresponding condition for non-linear Schrodinger equation.

Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile

An exact single envelope soliton solution of the damped nonlinear Schrödinger equation in a medium with a parabolic density profile has been obtained. It is found that the condition of reflection of the soliton from the density hump is determined by its initial velocity and the scale lengths of the inhomogeneity.

Envelope solitons and holes for sine-Gordon and non-linear Klein-Gordon equations

The non-linear Schrodinger equations governing the evolution of the plane wave solutions of the sine-Gordon and the non-linear Klein-Gordon equations are derived. The former is unstable against modulational instability and thus evolves into envelope solitons. The latter however can be unstable or stable depending on the sign of the cubic term and can accordingly admit envelope soliton or hole solutions.

Exact envelope-soliton solutions of a nonlinear wave equation

Exact N-envelope-soliton solutions have been obtained for the following nonlinear wave equation, i∂ψ/∂t + i3α|ψ|2 ∂ψ/∂x + β∂2ψ/∂x2 + iγ∂3ψ/∂x3 + δ|ψ|2ψ = 0, where α, β, γ and δ are real positive constants with the relation αβ = γδ. In one limit of α = γ = 0, the equation reduces to the nonlinear Schrödinger equation which describes a plane self-focusing and one-dimensional self-modulation of waves in nonlinear dispersive media. In another limit, β = δ = 0, the equation for real Ψ, reduces to the modified Korteweg-de Vries equation. Hence, the solutions reveal the close relation between classical solitons and envelope-solitons.