Interactions of breathers and solitons of a generalized variable-coefficient Korteweg-de Vries-modified Korteweg-de Vries equation with symbolic computation

Under investigation in this paper is a generalized variable-coefficient Korteweg-de Vries-modified Korteweg-de Vries equation which describes certain atmospheric blocking phenomenon. Lax pair and infinitely many conservation laws are obtained. With the help of the Hirota method and symbolic computation, the one-, two- and three-soliton solutions are given. Besides, breather and double pole solutions are derived. Propagation characteristics and interactions of breathers and solitons are discussed analytically and graphically. Results also show that the soliton changes its type between depression and elevation periodically. Parabolic-like breather and double pole are depicted. Conditions of the depression and elevation solitons are also given.     

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the combination of the third- and fifth-order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are close to solitons of the modified Korteweg-de Vries equation. However, the height of large-amplitude solutions has a limit approaching which solitary waves widen and acquire a table like shape similar to soluitons of the Gardner equation. Numerical calculations confirm that the collision of solitons of the derived equation is inelastic. Inelasticity is the most pronounced in the interaction of unipolar pulses. The direction of the shift of the phase of the higher-amplitude soliton owing to the interaction of solitons of different polarities depends on the amplitudes of the pulses.     

Generation of large-amplitude solitons in the extended Korteweg-de Vries equation

We study the extended Korteweg-de Vries equation, that is, the usual Korteweg-de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg-de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude "table-top" solitons as well as small-amplitude solitons similar to the solitons of the Korteweg-de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a "sech"-profile. (c) 2002 American Institute of Physics.     

Numerical simulation of the modified Korteweg-de Vries equation

The modified Korteweg-deVries (MKdV) is numerically solved using a new algorithm based on the finite element approach applying Galerkin’s method with quadratic spline interpolation functions. The stability of the proposed scheme is discussed. Numerical tests for one, two, and three solitons have been used to assess the performance of the proposed scheme.     

The perturbed Korteweg-de Vries equation: Evolution of solitons

Summary In this paper we examine the dynamic of solitons in the presence of external forces expressed by an-degree polynomial perturbative term or by a combination of polynomial and differential terms in the dependent variable. Under the action of these forces the soliton profile will no longer be a simple translation: asymptotic behaviour of the wave amplitude to threshold values (stationary equilibrium states) are now possible and “explosions” may occur at some finite “critical time” at which the soliton amplitude becomes infinite.     

Reflection of modified Korteweg-de Vries solitons in a plasmahaving negative ions

The variable coefficient modified Korteweg-de Vries (mKdV) equations for incident and reflected solitons are derived and solved to study the reflection of compressive and rarefactive ion acoustic solitons at the critical density in an inhomogeneous negative ion plasma. The polarity of the incident compressive and rarefactive solitons is not altered during the reflection process. Increasing the density gradient reinforces the reflection of both compressive and rarefactive mKdV solitons, whereas enhancement of the unperturbed plasma density weakens the reflection     

Interaction properties of complex modified Korteweg-de Vries (mKdV) solitons

Interaction properties of complex solitons are studied for the two U(1)-invariant integrable generalizations of the modified Korteweg-de Vries (mKdV) equation, given by the Hirota equation and the Sasa-Satsuma equation, which share the same traveling wave (single-soliton) solution having a sech profile characterized by a constant speed and a constant phase angle. For both equations, nonlinear interactions in which a fast soliton collides with a slow soliton are shown to be described by 2-soliton solutions that can have three different types of interaction profile, depending on the speed ratio and the relative phase angle of the individual solitons. In all cases, the shapes and speeds of the solitons are found to be preserved apart from a shift in position such that their center of momentum moves at a constant speed. Moreover, for the Hirota equation, the phase angles of the fast and slow solitons are found to remain unchanged, while, for the Sasa-Satsuma equation, the phase angles are shown to undergo a shift such that the relative phase between the fast and slow solitons changes sign.     

Residual symmetries of the modified Korteweg-de Vries equation and its localization

The residual symmetries of the famous modified Korteweg-de Vries (mKdV) equation are researched in this paper. The initial problem on the residual symmetry of the mKdV equation is researched. The residual symmetries for the mKdV equation are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries by means of enlarging the mKdV equations. One-parameter invariant subgroups and the invariant solutions for the extended system are listed. Eight types of similarity solutions and the reduction equations are demonstrated. It is noted that we researched the twofold residual symmetries by means of taking the mKdV equation as an example. Similarity solutions and the reduction equations are demonstrated for the extended mKdV equations related to the twofold residual symmetries.     

Breather-like solution of the Korteweg-de Vries equation

A new real singular solution of the Korteweg-de Vries equation is briefly described. It is shown that the spectrum of the associated linear problem consists of a pair of complex conjugate eigenvalues. The existence of constants of motion and eigenfunction normalization integral is also discussed.     

The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function g?₀ (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data u_x(0,t) and u_xx(0,t) are asymptotically t-periodic with period τ which tend to the functions g?₁ (t) and g?₂ (t) as t → ∞, respectively, then the periodic functions g?₁ (t) and g?₂ (t) can be uniquely determined in terms of the function g?₀ (t). Furthermore, we characterize the Fourier coefficients of g?₁ (t) and g?₂ (t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.     

On the relationship between the N-soliton solution of the modified Korteweg-de Vries equation and the KdV equation solution

The non-linear Miura transformation, which converts the N-soliton solution of the modified KdV equation into an N-soliton solution for the KdV equation itself, is related to an unitary transformation of the operators associated with these equations.     

The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vries equation

In this paper, a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation is investigated. With the help of symbolic computation, the N-soliton solution is derived through the Hirota method. Then the bilinear Bäcklund transformations and Lax pairs are presented. At last, we show some interactions of solitary waves.     

Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model

We consider the propagation of few-cycle pulses (FCPs) in cubic nonlinear media exhibiting a “crystal-like” structure, beyond the slowly varying envelope approximation, taking into account the wave polarization. By using the reductive perturbation method we derive from the Maxwell–Bloch–Heisenberg equations, in the long-wave-approximation regime, a non-integrable complex modified Korteweg-de Vries equation describing the propagation of circularly polarized (CP) FCPs. By direct numerical simulations of the governing nonlinear partial differential equation we get robust CP FCPs and we show that the unstable ones decays into linearly polarized half-cycle pulses, whose polarization direction slowly rotates around the propagation axis.     

Dispersion management for solitons in a Korteweg-de Vries system

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.     

Superconformal algebra and supersymmetric Korteweg-de Vries equation

It is shown that via the second hamiltonian structure, the superconformal algebra, realized in terms of Poisson brackets, is related to the unique (space) supersymmetric extension of the Korteweg-de Vries equation which is integrable. This generalizes a result obtained by Gervais and Neveu for the bosonic case.     

Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam

We address the problem of equipartition in a long Fermi-Pasta-Ulam (FPU) chain. After giving a precise relation between FPU and Korteweg-de Vries we use the latter equation to show that, corresponding to initial data a la Fermi, the time average of the energy on the kth mode decreases exponentially with kN. The result persists in the thermodynamic limit.     

Conserved densities of the fifth-order Korteweg-de Vries equation

It is proved that the rank of the non-trivial polynomial conserved density of the fifth-order KdV equation is 3p–2 or 3p (p=1, 2, ...).     

Nonlocal symmetries and similarity reductions for Korteweg-de Vries-negative-order Korteweg-de Vries equation

The nonlocal symmetries are derived for the Korteweg–de Vries–negative-order Korteweg–de Vries equation from the Painlevétruncation method.The nonlocal symmetries are localized to the classical Lie point symmetries for the enlarged system by introducing new dependent variables.The corresponding similarity reduction equations are obtained with different constant selections.Many explicit solutions for the integrable equation can be presented from the similarity reduction.