Semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site resonant coupling

We summarize the most featured items characterizing the semi-discrete nonlinear Schrödinger system with background-controlled inter-site resonant coupling. The system is shown to be integrable in the Lax sense that make it possible to obtain its soliton solutions in the framework of properly parameterized dressing procedure based on the Darboux transformation. On the other hand the system integrability inspires an infinite hierarchy of local conservation laws some of which were found explicitly in the framework of generalized recursive approach. The system consists of two basic dynamic subsystems and one concomitant subsystem and it permits the Hamiltonian formulation accompanied by the highly nonstandard Poisson structure. The nonzero background level of concomitant fields mediates the appearance of an additional type of inter-site resonant coupling and as a consequence it establishes the triangular-lattice-ribbon spatial arrangement of location sites for the basic field excitations. Adjusting the background parameter we are able to switch over the system dynamics between two essentially different regimes separated by the critical point. The system criticality against the background parameter is manifested both indirectly by the auxiliary linear spectral problem and directly by the nonlinear dynamical equations themselves. The physical implications of system criticality become evident after the rather sophisticated canonization procedure of basic field variables. There are two variants of system standardization equal in their rights. Each variant is realizable in the form of two nonequivalent canonical subsystems. The broken symmetry between canonical subsystems gives rise to the crossover effect in the nature of excited states. Thus in the under-critical region the system support the bright excitations in both subsystems, while in the over-critical region one of subsystems converts into the subsystem of dark excitations.     

The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method

A Bäcklund transformation both in bilinear form and in ordinary form for the transformed Vakhnenko equation is derived. An inverse scattering problem is formulated. The inverse scattering method has a third-order eigenvalue problem. A procedure for finding the exact N-soliton solution of the Vakhnenko equation via the inverse scattering method is described. The procedure is illustrated by considering the cases N=1 and N=2.     

Periodic and solitary-wave solutions of the Degasperis–Procesi equation

Travelling-wave solutions of the Degasperis–Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive x-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. A transformed version of the Degasperis–Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive x-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative x-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered.     

Nonlinear model of intramolecular excitations on a multileg ladder lattice

The exactly integrable model of nonlinear intramolecular excitations on a multileg ladder lattice is proposed. Since it is rather general, the model permits a number of physically interesting ramifications related to the striplike and the bunchlike biological and condensed matter systems as well as to the arrays of linearly and nonlinearly coupled optical fibers. The principal possibility to model an external magnetic field parallel to the ladder legs within the framework of inverse scattering transform is pointed out. The one-soliton solutions of two-leg and three-leg ladder models are found and analyzed. Apart from the spatially constricted translational mode typical to the traditional one-chain soliton, the interchain beating mode as well as the circular traveling modes redistributing the excitations between the chains are revealed in complete accordance with liner limits.     

Explicit solutions of the Camassa–Holm equation

Explicit travelling-wave solutions of the Camassa–Holm equation are sought. The solutions are characterized by two parameters. For propagation in the positive x-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis–Procesi equation are given in an appendix.     

Energy characteristics of a carbon monoxide gasdynamic laser

The characteristics of gasdynamic lasers based on mixtures of carbon monoxide with nitrogen and inert gases were investigated and the populations of vibrational levels of CO molecules, the gain of the mixture, and the generation power were determined in [1–8]. But the parameters of a gasdynamic laser (GDL) in the optimum emission mode have not been determined up to now. The difficulties in calculating the optimum energy characteristics are connected with the complexity of the calculating model and the large number of parameters of the system. The energy characteristics of a CO gasdynamic laser are calculated and optimized in the present report on the basis of a simple model.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, No. 5, pp. 16–23, September–October, 1978.     

Similarity in stationary motions of gas and two-phase medium with incompressible component

In this paper we suggest the transformation between the equations for a perfect gas and the equations describing in one-velocity approach the two-phase medium with any volume occupied by the incompressible phase. It is proved that the motion of a two-phase medium in the transformed coordinate system is similar with certain accuracy to that of a perfect gas. It means that the solutions obtained for perfect gas can be used to solve wave problems for media with incompressible component. There is no necessity directly to solve the problem for medium with incompressible component, and it is only sufficient to transform the known solution of the similar problem for a homogeneous medium. Thus, the solutions of many hydrodynamic problems for multi-component media with incompressible phase can be obtained without solving the original set of equations. The scope for the suggested transformation is demonstrated by reference to the strong explosion in a two-phase medium.     

Six-component semi-discrete integrable nonlinear Schrödinger system

We suggest the six-component integrable nonlinear system on a quasi-one-dimensional lattice. Due to its symmetrical form, the general system permits a number of reductions; one of which treated as the semi-discrete integrable nonlinear Schrödinger system on a lattice with three structural elements in the unit cell is considered in considerable details. Besides six truly independent basic field variables, the system is characterized by four concomitant fields whose background values produce three additional types of inter-site resonant interactions between the basic fields. As a result, the system dynamics becomes associated with the highly nonstandard form of Poisson structure. The elementary Poisson brackets between all field variables are calculated and presented explicitly. The richness of system dynamics is demonstrated on the multi-component soliton solution written in terms of properly parameterized soliton characteristics.