Solitons in relativistic mean field models

Assuming that the nucleus can be treated as a perfect fluid we study the conditions for the formation and propagation of Korteweg–de Vries (KdV) solitons in nuclear matter. The KdV equation is obtained from the Euler and continuity equations in nonrelativistic hydrodynamics. The existence of these solitons depends on the nuclear equation of state, which, in our approach, comes from well-known relativistic mean field models. We reexamine early works on nuclear solitons, replacing the old equations of state by new ones, based on QHD and on its variants. Our analysis suggests that KdV solitons may indeed be formed in the nucleus with a width which, in some cases, can be smaller than one Fermi.     

Experiments on Ion-Acoustic Solitons in Plasmas Invited Review Article

Experiments on ion-acoustic solitons are reviewed. Theories and numerical simulations which are relevant to experimental results are also presented. The measured velocity and width of planar solitons are compared with the predictions of the Korteweg-deVries (KdV) equation which includes a finite ion temperature. The spatial evolution of compressive or rarefactive pulses is discussed. Cylindrical and spherical solitons are introduced together with a modified KdV equation and numerical results. Oblique collisions and their evolution, including overtaking and head-on collisions of two planar, cylindrical, or spherical solitons are described. Reflection, diffraction, and other topics related with ion-acoustic solitons are presented.     

Effects of positron density and temperature on ion acoustic dressed solitons in an electron–positron–ion plasma

Ion acoustic dressed solitons in a three component plasma consisting of cold ions, hot electrons and positrons are studied. Using reductive perturbation method, Korteweg–de Vries (KdV) equation and a linear inhomogeneous equation, governing respectively the evolution of first and second order potentials are derived for the system. Renormalization procedure of Kodama and Taniuti is used to obtain nonsecular solutions of these coupled equations. It is found that electron–positron–ion plasma system supports only compressive solitons. For a given amplitude of soliton on increasing the positron concentration, velocity of the KdV as well as dressed soliton increases. For any arbitrary values of soliton's amplitude and positron concentration, velocity of the dressed soliton is found to be larger than that of the KdV soliton. For small amplitude of solitons, the width of KdV as well as dressed soliton decreases as positron concentration increases and width of dressed soliton is found to be larger than that of the KdV soliton. However, for a large value of soliton's amplitude as concentration of positrons increases, instead of decreasing width of dressed soliton starts to increase.     

A unified explicit form of Bäcklund transformations for generalized hierarchies of KdV equations

It is proved that the auto-Bäcklund transformations for all generalized KdV equations of any order admit a unified and explicit form. The theorem of permutability is proved in a unified way, too. The coefficients of the whole hierarchies of KdV equations considered are functions oft. The solitons whose speeds depend ont have unified formulas, too. Thus, we obtain vibrating solitons, solitons having infinite collisions, etc.     

On the problem of periodicity and hidden solitons for the KdV model

In continuum limit, the Fermi-Pasta-Ulam lattice is modeled by a Korteweg-de Vries (KdV) equation. It is shown that the long-time behavior of a KdV soliton train emerging from a harmonic excitation has a regular periodicity of right- and left-going trajectories. In a soliton train not all the solitons are visible, the solitons with smaller amplitude are hidden and their influence is seen through the changes of phase shifts of larger solitons. In the case of an external harmonic force several resonance schemes are revealed where both visible and hidden solitons have important roles. The weak, moderate, strong, and dominating fields are distinguished and the corresponding solution types presented.     

New exact solutions to the generalized KdV equation with generalized evolution

In this paper, by using a transformation and an application of Fan subequation, we study a class of generalized Korteweg–de Vries (KdV) equation with generalized evolution. As a result, more types of exact solutions to the generalized KdV equation with generalized evolution are obtained, which include more general single-hump solitons, multihump solitons, kink solutions and Jacobian elliptic function solutions with double periods.     

Space-Curved Resonant Line Solitons in a Generalized(2 + 1)-Dimensional Fifth-Order KdV System

On the basis of N-soliton solutions, space-curved resonant line solitons are derived via a new constraint proposed here, for a generalized(2 + 1)-dimensional fifth-order KdV system. The dynamic properties of these new resonant line solitons are studied in detail. We then discuss the interaction between a resonance line soliton and a lump wave in greater detail. Our results highlight the distinctions between the generalized(2+1)-dimensional fifth-order KdV system and the classical type.     

Exact solutions for KdV system equations hierarchy

Exact solutions for KdV system equations hierarchy are obtained by using the inverse scattering transform. Exact solutions of isospectral KdV hierarchy, nonisospectral KdV hierarchies and τ$\tau$-equations related to the KdV spectral problem are obtained by reduction. The interaction of two solitons is investigated.     

Modified Korteweg-De Vries equation for spatially inhomogeneous plasmas

The problem of propagation of ion-acoustic KdV solitons in weakly, spatially inhomogeneous plasmas is considered taking into account self-consistently the zeroth-order velocity and the potential existing in the system due to the presence of the inhomogeneity. An explicit soliton solution of the modified KdV equation is obtained.     

Negative-order KdV equations in (3+1) dimensions by using the KdV recursion operator

We develop a variety of negative-order Korteweg-de Vries (KdV) equations in (3+1)-dimensions. The recursion operator of the KdV equation is used to derive these higher dimensional models. The new equations give distinct solitons structures and distinct dispersion relations as well. We also determine multiple soliton solutions for each derived model.     

Analytic Solutions to Forced KdV Equation

The Korteweg-de Vries equation with a forcing term is established by recent studies as a simple mathematical model of describing the physics of a shallow layer of fluid subject to external forcing. In the present paper, we study the analytic solutions to the KdV equation with forcing term by using Hirota＇s direct method. Several exact solutions are given as examples, from which one can see that the same type soliton solutions can be excited by different forced term.     

Two-Soliton Solutions and Interactions for the Generalized Complex Coupled Kortweg-de Vries Equations

Kortweg-de Vries (KdV)-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV(GCCKdV) equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonic collisions are presented: (i) two solitons merge and separate fromeach other periodically; (ii) two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; (iii) two solitons are fluctuant in the central region of the collision. Propagation features ofsolitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the α increasing. Amplitude of v increase with the α increasing and decrease with the β increasing.     

THE PHYSICAL MECHANISM OF COLLISION BETWEEN SOLITONS

An easy and general way to access more complex soliton phenomena is introduced in this paper. The collision process between two solitons of the KdV equation is investigated in great detail with this novel approach, which is different from the sophisticated method of inverse scattering transformation. A more physical and transparent picture describing the collision of solitons is presented.     

Complex solitary waves and soliton trains in KdV and mKdV equations

We demonstrate the existence of complex solitary wave and periodic solutions of theKorteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. The solutions ofthe KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under thesimultaneous actions of parity (??) and time-reversal (??) operations. The corresponding localized solitons arehydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishingintensity. The ????-odd complex soliton solution is shown to beiso-spectrally connected to the fundamental sech² solution through supersymmetry. Physically, thesecomplex solutions are analogous to the experimentally observed grey solitons of non-liner Schödinger equation, governing the dynamics of shallow waterwaves and hence may also find physical verification.     

KdV-type solitons in multicomponent relativistic plasmas

A reductive perturbation technique is used to derive modified Korteweg-deVries (KdV) equations with different degrees of isothermality in a plasma, in order to study the existence and behavior of ion-acoustic solitary wave propagation ingoing in a multicomponent relativistic plasma. The solutions of the KdV equations are obtained. It is found that the presence of multiple ions and electrons in the relativistic plasma causes a different behavior regarding the formation of solitons in plasmas.     

Ion acoustic soliton experiments in a plasma

This paper reviews recent laboratory experiments on ion acoustic solitons that are excited, propagate, and interact in a plasma. The solitons can be described with the Korteweg–deVries (KdV) equation, the modified Korteweg–deVries (mKdV) equation, or the Kadomtsev–Petviashvilli (KP) equation. The results should be applicable in non- linear optics experiments.     

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.     

Eigenfunctions and Eigenvalues for a Scalar Riemann–Hilbert Problem Associated to Inverse Scattering

A complete set of eigenfunctions is introduced within the Riemann–Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schr?dinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation. Received: 17 December 1998/ Accepted: 21 July 1999     

Ion-acoustic solitons in multispecies spatially inhomogeneous plasmas

Ion-acoustic solitons are investigated in the spatially inhomogeneous plasma having electrons-positrons and ions. The soliton characteristics are described by Korteweg-de Vries equation which has an additional term. The density and temperature of different species play an important role for the amplitude and width of the solitons. Numerical calculations show only the possibility of compressive solitons. Further, analytical results predict that the peak amplitude of soliton decreases with the decrease of density gradient. Soliton characteristics like peak amplitude and width are substantially different from those based on KdV theory for homogeneous plasmas     

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.