Decay Of Solitary Waves Research Articles

Decay of solitary waves of fractional Korteweg-de Vries type equations

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the 1-dimensional semi-linear fractional equations:|D|αu+u−f(u)=0, with α∈(0,2), a prescribed coefficient p⁎(α), and a non-linearity f(u)=|u|p−1u for p∈(1,p⁎(α)), or f(u)=up with an integer p∈[2;p⁎(α)). Asymptotic developments of order 1 at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion α and of the non-linearity p. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory.

The existence and decay of solitary waves for the Fornberg–Whitham equation

In this paper, we consider the Fornberg-Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in our proof. Besides, we also prove that the family of solitary solutions is orbital stable and decay exponentially when speed wave c is bigger than 1.

Exact solution of Boussinesq equations for propagation of nonlinear waves

In this paper, we consider two Boussinesq models that describe propagation of small-amplitude long water waves. Exact solutions of the classical Boussinesq equation that represent the interaction of wave packets and waves on solitons are found. We use the Hirota representation and computer algebra methods. Moreover, we find various solutions for one of the variants of the Boussinesq system. In particular, these solutions can be interpreted as the fusion and decay of solitary waves, as well as the interaction of more complex structures.

Exponential decay and symmetry of solitary waves to Degasperis-Procesi equation

We improve the decay argument by Bona and Li (1997) [5] for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady Degasperis-Procesi equation in the nonlocal formulation. In addition, we give a method which confirms the symmetry of solitary waves, including those of the maximum height. Finally, we discover how the symmetric structure is connected to the steady structure of solutions to the Degasperis-Procesi equation, and give a more intuitive proof for symmetric solutions to be traveling waves. The improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations.

Numerical and perturbative computations of solitary waves of the Benjamin–Ono equation with higher order nonlinearity using Christov rational basis functions

Computation of solitons of the cubically-nonlinear Benjamin–Ono equation is challenging. First, the equation contains the Hilbert transform, a nonlocal integral operator. Second, its solitary waves decay only as O(1/∣ x∣ 2). To solve the integro-differential equation for waves traveling at a phase speed c, we introduced the artificial homotopy H( u XX ) − c u + (1 − δ) u 2 + δu 3 = 0, δ ∈ [0, 1] and solved it in two ways. The first was continuation in the homotopy parameter δ, marching from the known Benjamin–Ono soliton for δ = 0 to the cubically-nonlinear soliton at δ = 1. The second strategy was to bypass continuation by numerically computing perturbation series in δ and forming Padé approximants to obtain a very accurate approximation at δ = 1. To further minimize computations, we derived an elementary theorem to reduce the two-parameter soliton family to a parameter-free function, the soliton symmetric about the origin with unit phase speed. Solitons for higher order Benjamin–Ono equations are also computed and compared to their Korteweg–deVries counterparts. All computations applied the pseudospectral method with a basis of rational orthogonal functions invented by Christov, which are eigenfunctions of the Hilbert transform.

Comparison of three spectral methods for the Benjamin–Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions

The Benjamin–Ono equation is especially challenging for numerical methods because (i) it contains the Hilbert transform, a nonlocal integral operator, and (ii) its solitary waves decay only as O(1/| x| 2). We compare three different spectral methods for solving this one-space-dimensional equation. The Fourier pseudospectral method is very fast through use of the Fast Fourier Transform (FFT), but requires domain truncation: replacement of the infinite interval by a large but finite domain. Such truncation is unnecessary for a rational basis, but it is simple to evaluate the Hilbert Transform only when the usual rational Chebyshev functions TB n ( x) are replaced by their cousins, the Christov functions; the FFT still applies. Radial basis functions (RBFs) are slow for a given number of grid points N because of the absence of a summation algorithm as fast as the FFT; because RBFs are meshless, however, very flexible grid adaptation is possible.

Dissipation of Nonlinear Wave in Inhomogeneous Dust Plasma Crystal

A Korteweg-de Vires-type (KdV-type) equation and a modified Nonlinear Schrödinger equation (NLSE) for the dust lattice wave (DLW) are derived in a weakly inhomogeneous dust plasma crystal. It seems that the amplitude and the velocity of the dust lattice solitary waves decay exponentially with increasing time in a dust lattice. The modulational instability of this dust lattice envelope waves is investigated as well. It is found that the waves are modulational stable under certain conditions. On the other hand, the waves are modulational unstable if the conditions are not satisfied.

Propagation of sech2-type solitary waves in higher-order KdV-type systems

Abstract Wave propagation in microstructured media is essentially influenced by nonlinear and dispersive effects. The simplest model governing these effects results in the Korteweg–de Vries (KdV) equation. In the present paper a KdV-type evolution equation, including the third- and fifth-order dispersive and the fourth-order nonlinear terms, is used for modelling the wave propagation in microstructured solids like martensitic–austenitic alloys. The model equation is solved numerically under localised initial conditions. Possible solution types are defined and discussed. The existence of a threshold is established. Below the threshold, the relatively small solitary waves decay in time. However, if the amplitude exceeds a certain threshold, i.e., the critical value, then such a solitary wave can propagate with nearly a constant speed and amplitude and consequently conserve the energy.

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 the existence, regularity and decay of solitary waves to a generalized Benjamin–Ono equation

in R2, where ; ? 0 and (− )1=2 is the operator de2ned by F((− )1=2u)( )=| |u( ). F or ˆ represent the Fourier transform. Eq. (1) describes the dynamics of three-dimensional, slightly nonlinear disturbances in boundary-layer shear ;ows (without the assumption of a di<erence in their scales along and across the ;ow), see [1,6]. The solitary waves of (1) are solutions of the form A(x; y; t) = v(x − ct; y) where c is the speed of the solitary wave. It seems that solitary waves play an important role in the evolution of (1). Such a solution must satisfy the equation

Stability and Evolution of Solitary Waves in Perturbed Generalized Nonlinear Schrödinger Equations

In this paper, we study the stability and evolution of solitary waves in perturbed generalized nonlinear Schrodinger (NLS) equations. Our method is based on the completeness of the bounded eigenstates of the associated linear operator in L2 space and a standard multiple-scale perturbation technique. Unlike the adiabatic perturbation method, our method details all instability mechanisms caused by perturbations of such equations and explicitly specifies when such instabilities will occur. In particular, our method uncovers the instability caused by bifurcation of nonzero discrete eigenvalues of the linearization operator. As an example, we consider the perturbed cubic-quintic NLS equation in detail and determine the stability regions of its solitary waves. In the instability region, we also specify where the solitary waves decay, collapse, develop movingfronts, or evolve into a stable spatially localized and temporally periodic state. The generalization of this method to other perturbed nonlinear wave systems is also discussed.

Decay and analyticity of solitary waves

Considered here are detailed aspects of solitary-wave solutions of nonlinear evolution equations including the Euler equations for the propagation of gravity waves on the surface of an ideal, incompressible, inviscid fluid. Two properties will occupy our attention. The first, described already in an earlier paper, concerns the regularity of these travelling waves. In the context of certain classes of model equations for long waves in nonlinear dispersive media, we showed that solitary waves are obtained as the restriction to the real axis of functions analytic in a strip of the form { z : −a < s ( z) < a} in the complex plane. In this direction, the scope of our previous discussion of model equations is broadened considerably. Moreover, it is also shown that solitary-wave solutions of the full Euler equations have the properties that the free surface is given by the restriction to the real axis of a function analytic in a strip in the complex plane and the velocity potential is the restriction to the flow domain of a function that is analytic in an open set in complex 2-space C 2. The second issue considered is the asymptotic decay of solitary waves to a quiescent state away from their principal elevation. A theorem pertaining to the evanescence of solutions of certain types of one-dimensional convolution equations is formulated and proved, showing that decay is related to the smoothness of the Fourier transform of the convolution kernel k, as well as the nonlinearity present in the equation. It is demonstrated that if the Fourier transform k /_^ ϵ H s for some s > 1 2 , the rate of decay of a solution is at least as fast as that of the kernel k itself. This result is used to establish asymptotic properties of solitary-wave solutions of a broad class of model equations, and of solitary-wave solutions of the full Euler equations.

Space-charge solitary waves in charged particle beams in resistive-wall transport channels

The properties of solitary waves in space-charge dominated charged particle beams are investigated theoretically. Similarly to linear space-charge waves, both slow and fast solitary waves can exist. In a dissipative environment, e.g. a resistive-wall channel, it is shown that the slow space-charge solitary waves grow in amplitude, while the fast space-charge solitary waves decay. However, the growth and decay rates differ from those of the linear waves. A possible experiment with an electron beam is proposed to demonstrate these phenomena.

Instability and decay of solitary waves in the Davey-Stewartson 1 system

The exact solutions to the Davey-Stewartson 1 equations are constructed to describe the development of a transversal solitary wave instability. Two types of plane dark soliton decays into two-dimensional solitary waves are found and analysed. It is also shown that the transversal instability of one-dimensional waveguides leads only to a shift of their drift velocity under the action of nonlocalized, inhomogeneous perturbations.

Instability and decay of solitary waves in the Davey-Stewartson 1 system

The exact solutions to the Davey-Stewartson 1 equations are constructed to describe the development of a transversal solitary wave instability. Two types of plane dark soliton decays into two-dimensional solitary waves are found and analysed. It is also shown that the transversal instability of one-dimensional waveguides leads only to a shift of their drift velocity under the action of nonlocalized, inhomogeneous perturbations.

Energy propagation in dissipative systems. Part II: Centrovelocity for nonlinear wave equations

We consider nonlinear wave equations, first order in time, of a specific form. In the absence of dissipation, these equations are given by a Poisson system, with a Hamiltonian that is the integral of some density. The functional I, defined to be the integral of the square of the waveform, is a constant of motion of the unperturbed system. It will be shown that Z, the center of gravity of this density, is canonically conjugate to I and can be used as a measure to locate the position of the waveform. By introducing new coordinates based on Z, we get expressions for the centrovelocity, · Z , and the decay of I, which compare to the ones of part I, in both the conservative and the dissipative case. With the derived expressions, we investigate the decay of solitary waves of the Korteweg-de Vries equation, when different kinds of dissipation are added.

Numerical, experimental, and analytical studies of the dissipative toda lattice II. Elastic scattering of solitary waves

The behaviour of solitary waves in a strongly dissipative Toda lattice has been studied numerically and experimentally. Without dissipation the Toda lattice is an integrable soliton system, but due to dissipation the solitary waves decay and a tail is formed behind them. Numerical simulations of the head-on and the overtaking collisions and of the decomposition of the initial state indicate that the scattering is nevertheless elastic (within numerical accuracy). Experiments on a nonlinear electrical transmission line show the same features.

The rate of decay of solitary waves of finite amplitude

This paper is concerned with the problem of a solitary wave moving with constant form and constant velocity c on the surface of an incompressible, inviseid fluid over a horizontal bottom. The metion is assumed to be two-dimensional and irrotational, and if h is the depth of the fluid at infinity and g the acceleration due to gravity, then the Froude number F is defined by F2 = C2 /gh. It has been shown that necessarily F > 1, and Amick and Toland have recently conjectured that, if H(X) is the depth of the fluid at a horizontal distance x from the crest, then where ∝ is the smallest positive root of the equation and A is a positive number. The object of the paper is to prove this conjecture.