A Linear Energy-Preserving Finite Volume Element Method for the Improved Korteweg−de Vries Equation

In this paper, we introduce a linearized energy-preserving scheme which preserves the discrete global energy of solutions to the improved Korteweg?deVries equation. The method presented is based on the finite volume element method, by resorting to the variational derivative to transform the improved Korteweg?deVries equation into a new form, and then designing energy-preserving schemes for the transformed equation. The proposed scheme is much more efficient than the standard nonlinear scheme and has good stability. To illustrate its efficiency and conservative properties, we also compare it with other nonlinear schemes. Finally, we verify the efficiency and conservative properties through numerical simulations.     

Multi-hump bright solitons in a Schrödinger–mKdV system

We consider the problem of energy transport in a Davydov model along an anharmonic crystal medium obeying quartic longitudinal interactions corresponding to rigid interacting particles. The Zabusky and Kruskal unidirectional continuum limit of the original discrete equations reduces, in the long wave approximation, to a coupled system between the linear Schrödinger (LS) equation and the modified Korteweg–de Vries (mKdV) equation. Single- and two-hump bright soliton solutions for this LS–mKdV system are predicted to exist by variational means and numerically confirmed. The one-hump bright solitons are found to be the anharmonic supersonic analogue of the Davydov's solitons while the two-hump (in both components) bright solitons are found to be a novel type of soliton consisting of a two-soliton solution of mKdV trapped by the wave function associated to the LS equation. This two-hump soliton solution, as a two component solution, represents a new class of polaron solution to be contrasted with the two-soliton interaction phenomena from soliton theory, as revealed by a variational approach and direct numerical results for the two-soliton solution.     

Soliton–antisoliton collision in the ultradiscrete modified KdV equation

The discrete modified Korteweg–de Vries equation admits exact solutions with nondefinite sign, which describe interaction among solitons with positive and negative amplitude. In this Letter a transformation of hyperbolic sine type is proposed in order to ultradiscretize this equation and solutions.     

Helical solitons in vector modified Korteweg–de Vries equations

We study existence of helical solitons in the vector modified Korteweg–de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is confirmed by direct numerical simulations.     

Matrix KP: tropical limit and Yang–Baxter maps

We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of “pure line soliton solutions” for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang–Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a nonlinear map in the case of a more general matrix KP equation. We also consider the corresponding Korteweg–deVries reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain an apparently new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter dependence of the vector KP R-matrix.

     

The effect of the negative particle velocity in a soliton gas within Korteweg–de Vries-type equations

The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations (KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in [1], within the KdV equation. In the recent paper [2], an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.     

尘埃等离子体中非线性波的叠加效应及稳定性问题

研究了小的有限振幅的无磁场尘埃等离子体中的非线性波.在一维情况下由Kortewegde Veries(KdV)方程来描述,考虑了二维情况下尘埃等离子体中尘埃颗粒上电荷的变化效应以及双温度离子效应后,尘埃等离子体受到横向高阶扰动后动力学方程由Kadomtsev-Petviashvili(KP)方程来描述.在此基础上,研究了以任意夹角传播的两个及三个孤立子的相互作用问题,考虑非线性效应后振幅相等的双孤立子在相互作用区域内振幅最大值是单个孤立子振幅的4倍,振幅相等的三孤立子在相互作用区域内振幅最大值是单个孤立子振幅的9倍.研究还表明波的传播方向受到横向高阶扰动后是稳定的.     

Fairly-long water waves

Water waves have fascinated artists and poets, fishermen and surfers, as well as mathematicians, scientists, and engineers. Recent developments in the understanding of some types of water waves are cast in terms sufficiently general that they can be applied to waves in media other than water, for example, in high temperature plasmas, in gases, and in solids, as well. Historically, many important advances in wave theory have been made with water waves motivating the research. For example, the work of Airy (1845) on long nonlinear water waves preceded the partially analogous work on nonlinear sound waves of Riemann (1858–9) and Earnshaw (1858). The research of Kelvin (Thomson, 1887) leading to the enunciation of the method of stationary phase, which has found wide application to problems of wave propagation in any dispersive media, is entitled, On the Waves Produced by a Single Impulse in Water of any Depth, or in a Dispersive Medium. More recently, the properties of resonant interactions among waves in dispersive systems are known best for water waves (Phillips, 1960; Longuet-Higgins, 1962; Longuet-Higgins and Phillips, 1962; Benney, 1962; Hasselman, 1962; 1963a, b; McGoldrick, 1965), where experiments can be performed relatively easily, (McGoldrick et al., 1966). Some contemporary works in resonant interactions among other types of waves include waves in plasmas (Fishman et al., 1960; Litvak, 1960), electromagnetic waves in solids (Armstrong et al., 1962), and many others. Solid state scientists should note that the fundamental paper on heat conduction in solids by Peierls (1929) was the direct ancestor to the work over the last decade on resonant wave-wave interactions. Finally, remarkable properties of the Korteweg deVries Equation are only now being brought to light, the equation having originally been derived for water waves (Korteweg and deVries, 1895), but applicable to a wide variety of physical situations. The hope of the author is that the methods and results of modern research on water waves can find application in other fields of science. Broadly speaking, if a problem in waves involves dispersion and/or nonlinearity, chances are better that a solution exists, or has been attempted, for water waves than for any other type.     

Experimental observations of the dimensionless scaling of planar and cylindrical ion acoustic solitons

An experimental comparison of planar and cylindrical ion acoustic solitons is made through the use of dimensionless scaling parameters. The Korteweg de Vries equation is found to be insufficient to completely explain the observed data.     

Closed loop solitons and sigma functions: classical and quantized elasticas with genera one and two

Closed loop solitons in a plane, whose curvatures obey the modified Korteweg–de Vries equation, were investigated. It was shown that their tangential vectors are expressed by ratio of Weierstrass sigma functions for genus one case and ratio of Baker’s sigma functions for the genus two case. This study is closely related to classical and quantized elastica problems.     

Exact solutions for nonlinear variants of Kadomtsev–Petviashvili (n, n) equation using functional variable method

Studying compactons, solitons, solitary patterns and periodic solutions is important in nonlinear phenomena. In this paper we study nonlinear variants of the Kadomtsev–Petviashvili (KP) and the Korteweg–de Vries (KdV) equations with positive and negative exponents. The functional variable method is used to establish compactons, solitons, solitary patterns and periodic solutions for these variants. This method is a powerful tool for searching exact travelling solutions in closed form.     

Some features of one-soliton solutions of the Korteweg - de Vries equation

Using the method of inverse scattering problem [1, 2], we study solutions of the Korteweg - de Vries equation under initial conditions in the form of two nonsoliton pulses with not very large amplitudes. It is shown that if the distance between these pulses is not large, then they evolve to one soliton and an oscillating nonlinear tail for t → ∞. As the distance between the pulses or the pulse amplitudes increase, two solitons and an oscillating nonlinear tail are formed. Similar behavior is observed for solutions of the nonlinear Schrödinger equation. The only difference is that three, but not two, solitons are formed if the distance between two initial inphase pulses increases. The results of analytical consideration are illustrated by the numerical solution of the Korteweg - de Vries equation.     

Formation of freak waves in a soliton gas described by the modified Korteweg–de Vries equation

The nonlinear dynamics of multisoliton, differently polar fields is investigated within the framework of the modified Korteweg–de Vries equation. It is shown that the occurrence of abnormally large waves (freak waves) is possible in similar fields, which is associated with the modulation instability of cnoidal waves. The statistical moments of wave fields are investigated. It is shown that an increase in the coefficient of excess due to the interaction of solitons correlates with an increase in the probability of occurrence of freak waves. It is shown that the nonlinear interaction of differently polar solitons results in variation of the distribution functions of peak characteristics: the fraction of low-amplitude waves decreases, while that of the waves with large amplitudes increases. The dependence of the intensity of the density of the characteristics of the soliton gas is shown.     

Soliton Perturbation Theory for the Compound KdV Equation

The soliton perturbation theory is used to study the solitons that are governed by the compound Korteweg de-Vries equation in presence of perturbation terms. The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained. AMS Codes: 35Q51; 35Q53; 37K10. PACS Codes: 02.30.Jr; 02.30.Ik.     

A new family of four-dimensional symplectic and integrable mappings

We investigate the generalisations of the Quispel, Roberts and Thompson (QRT) family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables. We assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with Poisson brackets of the Weyl type. All mappings satisfying these conditions are shown to be integrable either as four-dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double-discrete versions of the modified Korteweg–deVries (ΔΔMKdV) and sine-Gordon (ΔΔsG) equations or by reduction to two-dimensional mappings with one integral of the symmetric QRT family.     

Ion‐ and positron‐acoustic solitons in magnetized dusty plasma with q‐non‐extensive electron and positron velocity distribution

In this study, the properties of ion‐ and positron‐acoustic solitons are investigated in a magnetized multi‐component plasma system consisting of warm fluid ions, warm fluid positrons, q‐non‐extensive distributed positrons, q‐non‐extensive distributed electrons, and immobile dust particles. To drive the Korteweg–de Vries (KdV) equation, the reductive perturbation method is used. The effects of the ratio of the density of positrons to ions, the temperature of the positrons, and ions to electrons, the non‐extensivity parameters q_e and q_p , and the angle of the propagation of the wave with the magnetic field on the potential of ion‐ and positron‐acoustic solitons are also studied. The present investigation is applicable to solitons in fusion plasmas in the edge of tokamak.     

Electrostatic Nonlinear Structures in Dissipative Electron--Positron--Ion Quantum Plasmas

Low frequency （in comparison to ion plasma frequency） ion-acoustic shocks and solitons in superdense electronpositron-ion quantum plasmas are studied. The quantum hydrodynamic model is used incorporating quantum Bohm forces and Fermi-Dirac statistical corrections to derive the deformed Korteweg de Vries-Burgers （dKdVB） equation in weakly nonlinear limit. The travelling wave solution of dKdVB equation is presented and results are discussed in different limits. It is found that shock height increases with increase of quantum pressure, positron concentration and dissipation. Further, it is seen that the width of soliton decreases with increase of quantum pressure     

Magnetoacoustic solitons in pair-ion fullerene plasma

ABSTRACT

The propagation of magnetoacoustic (fast magnetohydrodynamic) waves in pair-ion (PI) fullerene plasma is studied in the linear and nonlinear regimes. The pair-ion (PI) fullerene plasma is theorized as homogeneous, magnetized, warm and collisionless. Employing multi-fluid magnetohydrodynamic model, the dispersion relation is obtained and wave dispersion effects which appear through ion inertial length are discussed. Using reductive perturbation technique (RPT), the Korteweg–de Vries (KdV) equation is derived and its solution for small but finite amplitude magnetoacoustic solitons propagating in the direction perpendicular to the external magnetic field is presented. The compressive magnetoacoustic soliton (i.e. positive potential pulse) propagating with super Alfvénic speed is obtained in magnetized PI fullerene plasma. The variations in the amplitude and width of the magnetoacoustic soliton structures are also illustrated by using numerical values of the plasma parameters such as ions' density, temperature difference between fullerene ions and magnetic field intensity, which have been taken from the PI plasma experiments already published in the literature.     

Galerkin methods applied to some model equations for non-linear dispersive waves

The application of a dissipative Galerkin scheme to the numerical solution of the Korteweg de Vries (KdV) and Regularised Long Wave (RLW) equations, is investigated. The accuracy and stability of the proposed schemes is derived using a localised Fourier analysis. With cubic splines as basis functions, the errors in the numerical solutions of the KdV equation for different mesh-sizes and different amounts of dissipation is determined. It is shown that the Galerkin scheme for the RLW equation gives rise to much smaller errors (for a given mesh-size), and allows larger steps to be taken for the integrations in time (for a specified error tolerance). Also, the interaction of two solitons is compared for the KdV and RLW equations, and several differences in their behaviour are found.     

Engineering solitons and breathers in a deformed ferromagnet: Effect of localised inhomogeneities

We investigate the soliton dynamics of the electromagnetic wave propagating in an inhomogeneous or deformed ferromagnet. The dynamics of magnetization and the propagation of electromagnetic waves are governed by the Landau–Lifshitz–Maxwell (LLM) equation, a certain coupling between the Landau–Lifshitz and Maxwell's equations. In the framework of multiscale analysis, we obtain the perturbed integral modified KdV (PIMKdV) equation. Since the dynamic is governed by the nonlinear integro-differential equation, we rely on numerical simulations to study the interaction of its mKdV solitons with various types of inhomogeneities. Apart from simple one soliton experiments with periodic or localised inhomogeneities, the numerical simulations revealed an interesting dynamical scenario where the collision of two solitons on a localised inhomogeneity create a bound state which then produces either two separated solitons or a mKdV breather.