Completely integrable models of nonlinear optics

The models of the nonlinear optics in which solitons appeared are considered. These models are of paramount importance in studies of nonlinear wave phenomena. The classical examples of phenomena of this kind are the self-focusing, self-induced transparency and parametric interaction of three waves. At present there are a number of theories based on completely integrable systems of equations, which are, both, generations of the original known models and new ones. The modified Korteweg-de Vries equation, the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equation. Sine-Gordon equation, the reduced Maxwell-Bloch equation. Hirota equation, the principal chiral field equations, and the equations of massive Thirring model are some soliton equations, which are usually to be found in nonlinear optics theory.     

On nonlinear stability of general undercompressive viscous shock waves

We study the nonlinear stability of general undercompressive viscous shock waves. Previously, the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation, using new pointwise methods. By examining time invariants associated with the linearized equations, we can now overcome a new difficulty in the general case, namely, nonlinear movement of the shock. We introduce a coordinate transformation suitable to treat this new aspect, and demonstrate our method by analyzing a model system of generic type. We obtain sharp pointwise bounds andL ^p behavior of the solution for allp, 1p.     

On the solutions in elliptic functions of integrable nonlinear equations

A new solution in elliptic functions for the KdV equation is constructed using the method proposed by Belokolos and the author.     

On the stability of localized states in nonlinear field models with Lifshitz invariants

The static solutions for a nonlinear vector field in models with Lifshitz invariants are investigated. It is shown that in such systems two-and three-dimensional localized states, associated with the relaxation of the modulus of the vector field, are radially unstable. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 4, 296–298 (25 August 1998)     

Constraints on the currents in principal models giving rise to integrable nonlinear systems

Quadratic constraints on the current in the principal SU(2n) model stand at the origin of an extended reduction mechanism. We derive local conservation laws both for the reduced model and for a class of solutions of the principal SU(2n) model.     

Nonlocal models for nonlinear, dispersive waves

Considered herein are model equations for the unidirectional propagation of small-amplitude, nonlinear, dispersive, long waves such as those governed by the classical Korteweg-de Vries equation. Of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial, so that the operator modelling dispersive effect is nonlocal. Particular cases in view here are the Benjamin-Ono equation and the intermediate long-wave equation which arise in internal-wave theory, and Smith's equation which governs certain types of continental-shelf waves.

The initial-value problem for these equations is shown to be globally well posed in the classical sense, including continuous dependence upon the initial data and, in certain cases upon the modelling of nonlinear and dispersive effects. Whilst the results are stated for the specific equations listed above, the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear, dispersive evolution equations. Particularly worthy of note is our theorem implying that solutions of the intermediate long-wave equation converge strongly to solutions of the Korteweg-de Vries equation, or to solutions of the Benjamin-Ono equation, in appropriate asymptotic limits.     

On the geometry of classically integrable two-dimensional non-linear sigma models

A master equation expressing the zero curvature representation of the equations of motion of a two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. Special attention is paid to those representations possessing a spectral parameter. Furthermore, a closer connection between integrability and T-duality transformations is emphasised. Finally, new integrable non-linear sigma models are found and all their corresponding Lax pairs depend on a spectral parameter.     

Bifurcation,stability and symmetry of nonlinear waves

The bifurcation of wave-like spatio-temporal structures due to a hard-mode instability at non-zero wave number is investigated for a simple class of driven systems in one space dimension. We find generically a bifurcation of two branches of waves, travelling waves and standing waves, characterized by nontrivial subgroups of the symmetry group of the system. If both branches are supercritical, the wave with the larger amplitude is found to be stable. In all other cases, both waves are unstable for small amplitudes. At the common boundary of the stability regions of the two wave types in parameter space we find a bifurcation of a branch of modulated waves involving two independent frequencies, connecting the branches of travelling waves and standing waves.Work supported by the Swiss National Science Foundation     

Neumann-like integrable models

A countable class of integrable dynamical systems, with four-dimensional phase space and conserved quantities in involution (Hn,In)$(H_n,I_n)$ are exhibited. For n=1$n=1$ we recover Neumann system on T^∗S²$T^{\ast }{S}^2$. All these systems are also integrable at the quantum level.     

Dyonic integrable models

A class of non-abelian affine Toda models arising from the axial gauged two-loop WZW model is presented. Their zero curvature representation is constructed in terms of a graded Kac–Moody algebra. It is shown that the discrete multivacua structure of the potential together with non-abelian nature of the zero grade subalgebra allows soliton solutions with non-trivial electric and topological charges. The dressing transformation is employed to explicitly construct one and two soliton solutions and their bound states in terms of the tau functions. A discussion of the classical spectra of such solutions and the time delays are given in detail.     

Distortion products and backward-traveling waves in nonlinear active models of the cochlea

This study explores the phenomenology of distortion products in nonlinear cochlear models, predicting their amplitude and phase along the basilar membrane. The existence of a backward-traveling wave at the distortion-product frequency, which has been recently questioned by experiments measuring the phase of basilar-membrane vibration, is discussed. The effect of different modeling choices is analyzed, including feed-forward asymmetry, micromechanical roughness, and breaking of scaling symmetry. The experimentally observed negative slope of basilar-membrane phase is predicted by numerical simulations of nonlinear cochlear models under a wide range of parameters and modeling choices. In active models, positive phase slopes are predicted by the quasi-linear analytical computations and by the fully nonlinear numerical simulations only if the distortion-product sources are localized apical to the observation point and if the stapes reflectivity is unrealistically small. The results of this study predict a negative phase slope whenever the source is distributed over a reasonably wide cochlear region and/or a reasonably high stapes reflectivity is assumed. Therefore, the above-mentioned experiments do not contradict "classical" models of cochlear mechanics and of distortion-product generation.     

Supersymmetry in boundary integrable models

Quantum integrable models that possess N = 2 supersymmetry are investigated on the half-space. Conformal perturbation theory is used to identify some N = 2 supersymmetric boundary integrable models, and the effective boundary Landau-Ginzburg formulations are constructed. It is found that N = 2 supersymmetry largely determines the boundary action in terms of the bulk, and in particular, the boundary bosonic potential is |W|², where W is the bulk superpotential. Supersymmetry is also investigated using the affine quantum group symmetry of exact scattering matrices, and the affine quantum group symmetry of boundary reflection matrices is analyzed both for supersymmetric and more general models. Some N = 2 supersymmetry preserving boundary reflection matrices are given, and their connection with the boundary Landau-Ginzburg actions is discussed.     

On the second-type nonlinear elastic waves in porous media

Nonlinear second-type (matrix) waves are studied with special emphasis on the formation of saw-tooth shock waves. Configurations of the elastic waves in specific cases of porous gas-saturated sedimentary are calculated.     

Diffusive Toda chains in models of nonlinear waves in active media

A class of models of autowaves in the form of nonlinear diffusion equations, which are closely related to the Liouville equation and two-dimensionalized Toda chains, is investigated. Exact solutions of these equations are constructed and analyzed. A simple method for constructing diffusive Toda chain models from known basic models is proposed. Zh. éksp. Teor. Fiz. 114, 1897–1914 (November 1998)     

Completely integrable models in quasicrystals

The general method of construction of integrable dynamical models in quasicrystals is presented in the paper. It is illustrated on the example of the model of interacting spins for Penrose nonperiodic tiling of the plane. Another example constructed is the three dimensional model of interacting spins for icosahedral tiling of the three dimensional space. The bulk free energy is calculated for these models.     

On baxterized solutions of reflection equation and integrable chain models

Nonpolynomial baxterized solutions of reflection equations associated with affine Hecke and affine Birman–Murakami–Wenzl algebras are found. Relations to integrable spin chain models with nontrivial boundary conditions are discussed.     

Multicharged dyonic integrable models

We introduce and study new integrable models (IMs) of A_n⁽¹⁾-nonabelian Toda type which admit U(1)⊗U(1) charged topological solitons. They correspond to the symmetry breaking SU(n+1)→SU(2)⊗SU(2)⊗U(1)ⁿ⁻² and are conjectured to describe charged dyonic domain walls of N=1 SU(n+1) SUSY gauge theory in large n limit. It is shown that this family of relativistic IMs corresponds to the first negative grade q=−1 member of a dyonic hierarchy of generalized cKP type. The explicit relation between the 1-soliton solutions (and the conserved charges as well) of the IMs of grades q=−1 and q=2 is found. The properties of the IMs corresponding to more general symmetry breaking SU(n+1)→SU(2)^⊗p⊗U(1)^n−p as well as IM with global SU(2) symmetries are discussed.     

On classicalization in nonlinear sigma models

We consider the phenomenon of classicalization in nonlinear sigma models with both positive and negative target space curvature and with any number of derivatives. We find that the theories with only two derivatives exhibit a weak form of classicalization, and that the quantitative results depend on the sign of the curvature. Nonlinear sigma models with higher derivatives show a strong form of the phenomenon which is independent of the sign of curvature. We argue that weak classicalization may actually be equivalent to asymptotic safety, whereas strong classicalization seems to be a genuinely different phenomenon. We also discuss possible ambiguities in the definition of the classical limit.     

On modulation instability of nonlinear Alfven waves

The modulation instability of nonlinear Alfven waves in high-temperature plasma with _{e i} = 1 with allowance for resonance particles is discussed. A dispersion equation is derived for longitudinal perturbations and instability increments in regions æ << æ ₀ ^* , æ >> æ ₀ ^* , and æ æ ₀ ^* . For short-wave perturbations æ >> æ ₀ ^* , the dispersion equation leads to the well-known formula for the Landau coefficient of nonlinear decay. As an example of application of the obtained formulas, the correlation and cross-correlation functions are calculated for perturbations of the transverse magnetic-field component of nonlinear Alfven waves propagated along the magnetic field.Chechno-Ingushskii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 35, No. 8, pp. 663–670, August, 1992.