Quantization of spin direction for solitary waves in a uniform magnetic field

It is known that there are nonlinear wave equations with localized solitary wave solutions. Some of these solitary waves are stable (with respect to a small perturbation of initial data) and have nonzero spin (nonzero intrinsic angular momentum in the center of momentum frame). In this paper we consider vector-valued solitary wave solutions to a nonlinear Klein-Gordon equation and investigate the behavior of these spinning solitary waves under the influence of an externally imposed uniform magnetic field. We find that the only stationary spinning solitary wave solutions have spin parallel or anti-parallel to the magnetic field direction.     

Nonlinear dynamics of a continuum Heisenberg spin chain with an anisotropy and an external magnetic field

By stereographically projecting the spin vector onto a complex plane in the equations of motion for a continuum Heisenberg spin chain with an anisotropy (an easy plane and an easy axis) and an external magnetic field, the effect of the magnetic field for integrability of the system is discussed. Then, introducing an auxiliary parameter, the Lax equations for Darboux matrices are generated recursively. By choosing the constants, the Jost solutions are satisfied the corresponding Lax equations. The exact soliton solutions are investigated, then the total magnetic momentum and its z-component are obtained. These results show that the solitary waves depend essentially on two velocities which describe a spin configuration deviating from a homogeneous magnetization. The depths and widths of solitary waves vary periodically with time. The center of an inhomogeneity moves with a constant velocity, while the shape of soliton also changes with another velocity and this shape is not symmetrical with respect to the center. The total magnetic momentum and its z-component vary with time.     

On the dynamics of a continuum spin system

For a one-dimensional system of classical spins with nearest neighbour Heisenberg interaction we derive the equation of motion for each three-dimensional spin vector. In the continuum limit where the spins lie dense on a line this set of equations reduces to a nonlinear partial differential equation. In addition to spin-wave solutions we obtain some other special solutions of this equation. In particular we find solitary waves having total energy localised in a finite region, with velocity of propagation inversely proportional to the width of this region. Solutions of still another type are shown to have a diffusive character. The stability of such solutions and the possibility of interaction of two or more solitary waves have not yet been studied.     

Solitary waves on finite-size antiferromagnetic quantum Heisenberg spin rings

Motivated by the successful synthesis of several molecular quantum spin rings we are investigating whether such systems can host magnetic solitary waves. The small size of these spin systems forbids the application of a classical or continuum limit. We therefore investigate whether the time-dependent Schrödinger equation itself permits solitary waves. Example solutions are obtained via complete diagonalization of the underlying Heisenberg Hamiltonian.     

Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α. Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied.     

Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons

The properties of obliquely propagating ion-acoustic waves have been investigated in multi-ions magnetized plasma comprising of inertial, positively and negatively charged ion fluids, trapped electrons, and negatively charged stationary heavy ions. The propagation of the waves is oblique to the ambient magnetic field which is along the z-direction. Only fast type of modes exists in the linear regime. The reductive perturbation method was adopted to derive the Korteweg– de Vries (KdV) and Burger equations, as well as the solitary and shock wave solutions of the evolved equations, have been used to analyze the properties of the small but finite amplitude waves. The effects of the constituent plasma parameters, namely, the trapping effect of electrons, the electron degenerate temperature and the viscosity coefficient on the dynamics of the small amplitude solitary and shock waves have been examined. The influence of the magnetic field and the obliquity parameter on the propagation characteristics of ion-acoustic waves are discussed.     

Solitary wave complexes in two-component condensates

Axisymmetric three-dimensional solitary waves in uniform two-component mixture Bose-Einstein condensates are obtained as solutions of the coupled Gross-Pitaevskii equations with equal intracomponent but varying intercomponent interaction strengths. Several families of solitary wave complexes are found: (1) vortex rings of various radii in each of the components; (2) a vortex ring in one component coupled to a rarefaction solitary wave of the other component; (3) two coupled rarefaction waves; (4) either a vortex ring or a rarefaction pulse coupled to a localized disturbance of a very low momentum. The continuous families of such waves are shown in the momentum-energy plane for various values of the interaction strengths and the relative differences between the chemical potentials of two components. Solitary wave formation, their stability, and solitary wave complexes in two dimensions are discussed.     

Interfacial waves with free-surface boundary conditions: an approach via a model equation

In a two-fluid system where the lower fluid is bounded below by a rigid bottom and the upper fluid is bounded above by a free surface, two kinds of solitary waves can propagate along the interface and the free surface: classical solitary waves characterized by a solitary pulse or generalized solitary waves with nondecaying oscillations in their tails in addition to the solitary pulse. The classical solitary waves move faster than the generalized solitary waves. The origin of the nonlocal solitary waves can be understood from a physical point of view. The dispersion relation for the above system shows that short waves can propagate at the same speed as a “slow” solitary wave. The interaction between the solitary wave and the short waves creates a nonlocal solitary wave. In this paper, the interfacial-wave problem is reduced to a system of ordinary differential equations by using a classical perturbation method, which takes into consideration the possible resonance between short waves and “slow” solitary waves. In the past, classical Korteweg–de Vries type models have been derived but cannot deal with the resonance. All solutions of the new system of model equations, including classical as well as generalized solitary waves, are constructed. The domain of validity of the model is discussed as well. It is also shown that fronts connecting two conjugate states cannot occur for “fast” waves. For “slow” waves, fronts exist but they have ripples in their tails.     

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 Alfvén 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.     

Propagation and interaction of ion-acoustic solitary waves in a quantum electron-positron-ion plasma

This paper discusses the existence of ion-acoustic solitary waves and their interaction in a dense quantum electron-positron-ion plasma by using the quantum hydrodynamic equations.The extended Poincar’e-Lighthill-Kuo perturbation method is used to derive the Korteweg-de Vries equations for quantum ion-acoustic solitary waves in this plasma.The effects of the ratio of positrons to ions unperturbation number density p and the quantum diffraction parameter H e (H p) on the newly formed wave during interaction,and the phase shift of the colliding solitary waves are studied.It is found that the interaction between two solitary waves fits linear superposition principle and these plasma parameters have significantly influence on the newly formed wave and phase shift of the colliding solitary waves.The investigations should be useful for understanding the propagation and interaction of ion-acoustic solitary waves in dense astrophysical plasmas (such as white dwarfs) as well as in intense laser-solid matter interaction experiments.     

Temporal solitary waves due to optical rectification and the electro-optic effect

Temporal solitary waves that are due to the mutual interaction of optical rectification and the electro-optic effect in the presence of a second-order nonlinearity are studied. It is found that a two-parameter family of solitons exists. Analytical solutions are found for special cases. Numerical soliton solutions of the system of equations include single- and multiple-hump solitary waves.     

Nonlinear electrostatic waves in a magnetized plasma

The modulational interaction of finite-amplitude high-frequency electrostatic waves propagating at an arbitrary angle to an external magnetic field with slow plasma motion is considered. A set of nonlinear evolution equations describing the interaction is obtained. New types of solitary waves propagating at near sonic speeds are found.     

Spin waves and spin polarizability in atoms

Linearized hydrodynamic equations for spin-up and -down fluids oscillating about the Thomas-Fermi ground state are derived variationally and estimates for the lowest-lying spin waves obtained. Static solutions in an external magnetic field yield a model spin susceptibility.     

Domain-wall induced phase shifts in spin waves

We study the interaction between two important features of ferromagnetic nanoparticles: magnetic domain walls and spin waves. Micromagnetic simulations reveal that magnetostatic spin waves change their phase as they pass through domain walls. Similar to an Aharonov-Bohm experiment, we suggest to probe this effect by splitting the waves on different branches of a ring. The interference of merging waves depends on the domain walls in the branches. A controlled manipulation of spin-wave phases could be the first step towards nanoscaled ferromagnetic devices performing logical operations based on spin-wave propagation.     

Varieties of solitons and solitary waves

A summary is presented of the principal types of completely integrable partial differential equations having soliton solutions. Each type is derived from an appropriate physical model of an electromagnetic wave problem, with the intention to show how known mathematical results apply to a coherent class of physical problems in electromagnetic waves. The non-linear Schrödinger (NS) equation appears when the induced non-linear dielectric polarization is expanded in a series of powers of the electric field, only the linear and third-order polarizations are retained, and the temporal spectrum of the wave is a narrow band far removed from any resonance of the medium. The sine-Gordon equation appears from a similar optical model of propagation in a dielectric consisting of identical 2-level atomic systems, but resonance occurs between the carrier frequency of the wave and the transition frequency of the atoms. The Boussinesq and Korteweg– de Vries equations appear at different levels of approximation to a potential wave on a transmission line having a non-linear capacitance such that the charge stored is a non-linear function of the line potential. In all cases the evolution variable is the propagation distance; the transverse variable is time, but in the case of the NS equation it may alternatively be a spatial coordinate, giving rise to the possibility of spatial solitons as well as temporal solitons for NS-type problems. Two examples are derived of non-integrable Hamiltonian systems having spatial solitary waves, namely the second-order cascade interaction and vector spatial solitary waves of the third-order interaction, and a brief survey of the analytical solutions for the plane waves and solitary waves of these two types is presented. Finally, the addition of a second spatial dimension to the non-linear transmission line problem leads to the Kadomtsev–Petviashvili equations, and a further approximation for weakly modulated travelling waves leads to the Davey–Stewartson equations. Both of these completely integrable systems support combined spatial–temporal solitons.     

Random generation of coherent solitary waves from incoherent waves

The random generation of coherent solitary waves from incoherent waves in a medium with an instantaneous nonlinearity has been observed. One excites a propagating incoherent spin wave packet in a magnetic film strip and observes the random appearance of solitary wave pulses. These pulses are as coherent as traditional solitary waves, but with random timing and a random peak amplitude.     

Kinks in a nonlinear massive sigma model

We describe the kink solitary waves of a massive nonlinear sigma model with an S2 sphere as the target manifold. Our solutions form a moduli space of nonrelativistic solitary waves in the long wavelength limit of ferromagnetic linear spin chains.     

Stability of solitary waves in nonlinear composite media

The system of equations for planar waves in elastic composite media in the presence of anisotropy is considered. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In case of the absence of anisotropy these two families coalesce into the unique three parametric family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in an anisotropic as well as in an isotropic materials. It is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result. The local well-posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave. This complements the previous results of blow-up for this type of equations.     

Oblique magneto-acoustic solitons in a rotating plasma

Weakly nonlinear magneto-acoustic waves propagating at an arbitrary angle to the external magnetic field in a rotating plasma are considered. A model equation (Ostrovsky's equation with positive dispersion) is derived from a set of basic magneto-hydrodynamic equations. Stationary solutions of this equation are obtained numerically and analyzed in detail theoretically. These include solitary-type solutions (solitons with monotonic and oscillating tails), complex multisolitons (bound states of coupled single solitons), as well as periodic waves. We emphasize that the positive dispersion, in contrast to the negative one, gives rise to solitary waves within the framework of Ostrovsky's equation.