Types Of Soliton Solutions Research Articles

Nonlinear stationary whistler waves and whistler solitons (oscillitons). Exact solutions

A fully nonlinear theory for stationary whistler waves propagating parallel to the ambient magnetic field in a cold plasma has been developed. It is shown that in the wave frame proton dynamics must be included in a self-consistent manner. The complete system of nonlinear equations can be reduced to two coupled differential equations for the transverse electron or proton speed and its phase, and these possess a phase-portrait integral which provides the main features of the dynamics of the system. Exact analytical solutions are found in the approximation of ‘small’ (but nonlinear) amplitudes. A soliton-type solution with a core filled by smaller-scale oscillations (called ‘oscillitons’) is found. The dependence of the soliton amplitude on the Alfvén Mach number, and the critical soliton strength above which smooth soliton solutions cannot be constructed is also found. Another interesting class of solutions consisting of a sequence of wave packets exists and is invoked to explain observations of coherent wave emissions (e.g. ‘lion roars’) in space plasmas. Oscillitons and periodic wave packets propagating obliquely to the magnetic field also exist although in this case the system becomes much more complicated, being described by four coupled differential equations for the amplitudes and phases of the transverse motion of the electrons and protons.

Wave Regimes of Viscous Film Flow down a Vertical Cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.

Soliton solutions for quasilinear Schrödinger equations, II

For a class of quasilinear Schrödinger equations, we establish the existence of ground states of soliton-type solutions by a variational method.

Localized excitations in (2+1)-dimensional systems.

By means of a special variable separation approach, a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various (2+1)-dimensional models such as the Davey-Stewartson (DS) model, the Nizhnik-Novikov-Veselov (NNV) system, asymmetric NNV equation, asymmetric DS equation, dispersive long wave equation, Broer-Kaup-Kupershmidt system, long wave-short wave interaction model, Maccari system, and a general (N+M)-component Ablowitz-Kaup-Newell-Segur (AKNS) system. Selecting the arbitrary functions appropriately, one may obtain abundant stable localized interesting excitations such as the multidromions, lumps, ring soliton solutions, breathers, instantons, etc. It is shown that some types of lower dimensional chaotic patterns such as the chaotic-chaotic patterns, periodic-chaotic patterns, chaotic line soliton patterns, chaotic dromion patterns, fractal lump patterns, and fractal dromion patterns may be found in higher dimensional soliton systems. The interactions between the traveling ring type soliton solutions are completely elastic. The traveling ring solitons pass through each other and preserve their shapes, velocities, and phases. Some types of localized weak solutions, peakons, are also discussed. Especially, the interactions between two peakons are not completely elastic. After the interactions, the traveling peakons also pass through each other and preserve their velocities and phases, however, they completely exchange their shapes.

Soliton solutions for quasilinear Schrödinger equations, I

For a class of quasilinear Schrödinger equations we establish the existence of ground states of soliton type solutions by a minimization argument.

Solutions of a (2+1)-dimensional dispersive long wave equation.

A special type of multisoliton solution with a particular dispersion relation is obtained for Wu-Zhang equation [which describes (2+1)-dimensional dispersive long waves] by the standard Weiss-Tabor-Carnvale Painlevé truncation expansion. Using a nonstandard truncation of a modified Conte's invariant Painlevé expansion, two different types of soliton solutions without any dispersive relation is found. Two types of periodic wave solutions expressed by Jacobi elliptic functions are found by the truncations of a special extended Painlevé expansion. The soliton solutions are special cases of the corresponding periodic solutions.

Solitary waves, steepening and initial collapse in the Maxwell–Lorentz system

We present a numerical study of Maxwell’s equations in nonlinear dispersive optical media describing propagation of pulses in one Cartesian space dimension. Dispersion and nonlinearity are accounted for by a linear Lorentz model and an instantaneous Kerr nonlinearity, respectively. The dispersion relation reveals various asymptotic regimes such as Schrödinger and KdV branches. Existence of soliton-type solutions in the Schrödinger regime and light bullets containing few optical cycles together with dark solitons are illustrated numerically. Envelope collapse regimes of the Schrödinger equation are compared to the full system and an arrest mechanism is clearly identified when the spectral width of the initial pulse broadens beyond the applicability of the asymptotic behavior. We show that beyond a certain threshold the carrier wave steepens into an infinite gradient similarly to the canonical Majda–Rosales weakly dispersive system. The weak dispersion in general cannot prevent the wave breaking with instantaneous or delayed nonlinearities.

Dynamics of solitons in a three-level medium with linear birefringence

The dynamics of ultrashort pulses in a three-level or degenerate two-level medium is studied in the framework of integrable systems of the Maxwell-Bloch equations for pulses with a duration much longer than or comparable with the inverse transition frequency, with allowance made for linear birefringence. Using soliton-type solutions, we study the effect of birefringence on soliton conversion for different initial populations of the levels.

Soliton solutions in three linearly coupled Korteweg–de Vries equations

We demonstrate that a purely symmetric system of three linearly coupled Korteweg–de Vries equations, which describe the resonant interaction of internal waves in stratified fluids, possesses propagating solutions of soliton type. The amplitude equations of this system, which are derived using the multiple-scales analysis, have a similar form as the sum frequency generation equations system in diffractive medium; that have recently been the focus of much efforts. For the stationary case, the solitary wave solutions are determined in two different process of generation. In addition, we found that solutions determined in the non-stationary case are also of soliton type. A comparison is made with results obtained by more general theoretical and numerical studies reported in literature.

Algebraic geometrical solutions forcertain evolution equations and Hamiltonian flows on nonlinearsubvarieties of generalized Jacobians

Algebraic geometrical solutions of a new shallow-water equationand Dym-type equation are studied in connection with Hamiltonianflows on nonlinear subvarieties of hyperelliptic Jacobians.These equations belong to a class of N-component integrablesystems generated by Lax equations with energy-dependentSchrödinger operators having poles in the spectral parameter.The classes of quasi-periodic and soliton-type solutions ofthese equations are described in terms of theta- andtau-functions by using new parametrizations. A qualitativedescription of real-valued solutions is provided.

Self-gravitating fluid dynamics, instabilities, and solitons

This work studies the hydrodynamics of self-gravitating, compressible, isothermal fluids. We show that the hydrodynamic-evolution equations are scale covariant in the absence of viscosity. Then, we study the evolution of the time-dependent fluctuations around singular and regular isothermal spheres. We linearize the fluid equations around such stationary solutions and develop a method based on the Laplace transform to analyze their dynamical stability. We find that the system is stable below a critical size $(X\ensuremath{\sim}9.0$ in dimensionless variables) and unstable above; this criterion is the same as the one found for the thermodynamic stability in the canonical ensemble and it is associated with a center-to-border density ratio of 32.1. We prove that the value of this critical size is independent of the Reynolds number of the system. Furthermore, we give a detailed description of the series of successive dynamic instabilities that appear at larger and larger sizes following the geometric progression ${X}_{n}\ensuremath{\sim}{10.7}^{n},$ $n=1,2,\dots{}.$ Then, we search for exact solutions of the hydrodynamic equations without viscosity, we provide analytic and numerical axisymmetric soliton-type solutions. The stability of exact solutions corresponding to a collapsing filament is studied by computing linear fluctuations. Radial fluctuations growing faster than the background are found for all sizes of the system. However, a critical size $(X\ensuremath{\sim}4.5)$ appears, separating a weakly from a strongly unstable regime.

Symmetry analysis and exact solutions of the 2+1 dimensional sine-Gordon system

According to the space variable exchange symmetry, we change the 2+1 dimensional sine-Gordon system obtained by Konopelchenko and Rogers (KR) to a variant form. Some types of similarity reductions are obtained by using some Lie symmetry analysis. From these similarity reductions, we find that the soliton structure of the system possesses quite a rich structure. The line solitons parallel to the lines x+y=0 and x−y=0 may have an arbitrary shape. In addition to the well-known dromion solution, which is constructed by two line solitons, one may find many other kinds of soliton solutions localized in all directions. For instance, some kinds of ring type (basin-like, plateau-like and bowl-like) and instanton type soliton solutions, can be found directly by selecting the arbitrary functions included in the “single” soliton solution. The dromion solutions may also be constructed by straight line and curved line solitons.

Magnetoelastic Interaction in the Heisenberg Magnet Model

We investigate the Heisenberg magnetic chain of spins interacting with the atomic displacement by exchange interaction modulation. By use of the SU(2) generalized coherent states technique we obtain in continual limit the classical description of the model, which is a system of coupled Landau–Lifshitz and Boussinesq equations describing nonlinear spin waves accompanied with sound waves. Domain-walls and soliton type solutions of this system are derived in harmonic phonon approximation. The possibility of energy pumping from the magnon subsystem to the phonon one is shown for the case of near-sonic velocities of motion of the solitons. The model of the multisublattice Heisenberg magnet consisting of p sublattices of ferromagnetic type and q sublattices of antiferromagnetic type is studied taking into consideration the interaction between the sublattices. A system of equations describing this model, which in the small amplitude of spin deviations approximation can be reduced to the nonlinear Schrödinger equation with Yajima–Oikawa potential, is obtained.

Decay of solitons in an isotropic collisionless quasineutral plasma with isothermal pressure

Soliton-type solutions of the complete unreduced system of transport equations describing the plane-parallel motions of an isotropic collisionless quasineutral plasma in a magnetic field with constant ion and electron temperatures are studied. The regions of the physical parameters for fast and slow magnetosonic branches, where solitons and generalized solitary waves—nonlocal soliton structures in the form of a soliton “core” with asymptotic behavior at infinity in the form of a periodic low-amplitude wave—exist, are determined. In the range of parameters where solitons are replaced by generalized solitary waves, soliton-like disturbances are subjected to decay whose mechanisms are qualitatively different for slow and fast magnetosonic waves. A specific feature of the decay of such disturbances for fast magnetosonic waves is that the energy of the disturbance decreases primarily as a result of the quasistationary emission of a resonant periodic wave of the same nature. Similar disturbances in the form of a soliton core of a slow magnetosonic generalized solitary wave essentially do not emit resonant modes on the Alfven branch but they lose energy quite rapidly because of continuous emission of a slow magnetosonic wave. Possible types of shocks which are formed by two types of existing soliton solutions (solitons and generalized solitary waves) are examined in the context of such solutions.

Solitons in spiral polymeric macromolecules.

The problem of the existence and stability of dynamical soliton regimes in a helix polymer is solved numerically. For the polytetrafluoroethylene macromolecule, within a model in which deformations of the valence and torsion angles and the valence bonds are taken into account, two types of soliton solutions are found. The first type describes the propagation of a solitary wave of torsional displacements of a helix chain. The twisting of the chain is a result of the compression of dihedral (torsion) angles. The second type describes the propagation of a solitary wave of longitudinal displacements of a helix chain. The longitudinal compression of the chain is a result of the compression of the valence angles and bonds. The solitons have a finite narrow spectrum of supersonic velocities: the soliton of torsion has a spectrum above the velocity of long-wavelength phonons of torsion while the spectrum of the solitons of compression lies above the velocity of long-wavelength phonons of longitudinal displacement. Numerical simulations of the soliton dynamics show their stability in the intervals of admissible velocities. The elasticity of soliton interactions under their collisions is demonstrated. The formation of solitons induced by deformation of end bonds of the helix chain has been modeled. It is shown that helicity of the macromolecule is the necessary condition for existence of torsional solitons.

Graded Lie algebras, representation theory, integrable mappings, and integrable systems

A new class of integrable mappings and chains is introduced. The corresponding 1+2 integrable systems that are invariant under such integrable mappings are presented in an explicit form. Soliton-type solutions of these systems are constructed in terms of matrix elements of fundamental representations of semisimple An algebras for a given group element. The possibility of generalizing this construction to the multidimensional case is discussed.

On soliton-type solutions of equations associated with N-component systems

The algebraic geometric approach to N-component systems of nonlinear integrable PDE’s is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.

New Non-traveling Solitary Wave Solutions for a Second-order Korteweg-de Vries Equation

Abstract Modeling the propagation of two different wave modes simultaneously, the second-order KdV equation is of current interest. Applying a tanh-typed method with symbolic computation, we have found certain new analytic soliton-typed solutions which go beyond the the previously obtained traveling wave solutions.

Auto-Bäcklund transformation and analytic solutions for general variable-coefficient KdV equation

We find analytic solutions for the general variable-coefficient KdV equation of the form u t + f( t) uu x + g( t) u xxx =0. We make use of both the application of the truncate Painlevé expansion and symbolic computation to obtain an auto-Bäcklund transformation and certain soliton-typed analytic solutions with a constraint on f( t) and g( t).

Nonlinear whistler wave propagation in bi-Maxwellian plasmas

The coupling between electron whistlers and nonlinear ion-acoustic density fluctuations in a plasma with bi-Maxwellian distributed electrons is considered by including the combined effects of relativistic electron-mass variation and ponderomotive force nonlinearities. The coupled mode propagation is governed by a generalized system of Schrödinger-Boussinesq equations, which admits localized exact analytical solutions for stationary propagation. Parameter regimes for the existence of different types of soliton solutions are discussed.