APPROXIMATE TECHNIQUES FOR DISPERSIVE SHOCK WAVES IN NONLINEAR MEDIA

Dispersive shock waves dominate wave-breaking phenomena in Hamiltonian systems. In the absence of loss, these highly irregular and disordered waves are potentially reversible. However, no experimental evidence has been given about the possibility of inverting the dynamics of a dispersive shock wave and turn it into a regular wave-front. Nevertheless, the opposite scenario, i.e., a smooth wave generating turbulent dynamics is well studied and observed in experiments. Here we introduce a new theoretical formulation for the dynamics in a highly nonlocal and defocusing medium described by the nonlinear Schroedinger equation. Our theory unveils a mechanism that enhances the degree of irreversibility. This mechanism explains why a dispersive shock cannot be reversed in evolution even for an arbitrarirly small amount of loss. Our theory is based on the concept of nonlinear Gamow vectors, i.e., power dependent generalizations of the counter-intuitive and hereto elusive exponentially decaying states in Hamiltonian systems. We theoretically show that nonlinear Gamow vectors play a fundamental role in nonlinear Schroedinger models: they may be used as a generalized basis for describing the dynamics of the shock waves, and affect the degree of irreversibility of wave-breaking phenomena. Gamow vectors allow to analytically calculate the amount of breaking of time-reversal with a quantitative agreement with numerical solutions. We also show that a nonlocal nonlinear optical medium may act as a simulator for the experimental investigation of quantum irreversible models, as the reversed harmonic oscillator.

This paper explores the application of Whitham modulation theory to the third-order focusing Kaup–Newell model, offering a complete classification of solutions for step-like initial value problems. Using the finite-gap integration method, we derive periodic solutions and the corresponding Whitham modulation equations. By analyzing the distribution of Riemann invariants, we identify the fundamental wave structures emerging from the step-like initial value problem. Furthermore, we provide a complete classification of solutions for this problem. Our results demonstrate that Whitham modulation theory serves as an effective analytical framework for studying initial value discontinuities in the third-order focusing KN model, offering new insights into its nonlinear dynamical behavior. Moreover, the direct numerical simulations show remarkable agreement with the results from Whitham modulation theory.

The steady two-dimensional problem of reflexion of an oblique partly dispersed plane shock wave from a plane wall is studied analytically. Viscosity, diffusion and heat conduction are neglected. The thermodynamic state of the gas is assumed to be determined by the instantaneous values of the specific entropy s, pressure p and a finite number of internal state variables. Results for the flow field behind the reflected shock are obtained by a perturbation method which is based on the assumption that the influence of relaxation is relatively weak.

We consider dispersive shock waves of the focusing nonlinear Schrödinger equation generated by discontinuous initial conditions which are periodic or quasiperiodic on the left semiaxis and zero on the right semiaxis. As an initial function, we use a finite-gap potential of the Dirac operator given in an explicit form through hyperelliptic theta-functions. The aim of this paper is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in the work of Kotlyarov and Khruslov [Teor. Mat. Fiz. 68(2), 751–761 (1986)] and Kotlyarov {Mat. Zametki 49(2), 84–94 (1991) [Math. Notes 49(1-2), 172–180 (1991)]} using Marchenko’s inverse scattering techniques. We investigate this problem exceptionally using the Riemann-Hilbert (RH) problem techniques that allow us to obtain explicit formulas for asymptotic solitons themselves in contrast with the cited papers where asymptotic formulas are obtained only for the square of the absolute value of solution. Using transformations of the main RH problems, we arrive at a model problem corresponding to the parametrix at the end points of the continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials, which naturally appeared after appropriate scaling of the Riemann-Hilbert problem in small neighborhoods of the end points of the continuous spectrum. Further asymptotic analysis gives an explicit formula for solitons at the edge of dispersive waves. Thus, we give the complete description of the train of asymptotic solitons: not only bearing the envelope of each asymptotic soliton, but its oscillating structure is found explicitly. Besides, the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock waves.

Head-on collision and overtaking collision between a KdV solitary wave and an envelope solitary wave are first studied in present paper by using Particle-in-cell (PIC) method in a dusty plasma. There are phase shifts of the KdV solitary wave in both head-on collision and the overtaking collision, while no phase shift is found for the envelop solitary wave in any cases. The remarkable difference between head-on collision and the overtaking collision is that the phase shift of KdV solitary wave increases as amplitude of KdV solitary wave increases in head-on collision, while it decreases as amplitude of the KdV solitary wave increases in the overtaking collision. It is found that the maximum amplitude during the collision process is less than sum of two amplitudes of both solitary waves, but is larger than either of the amplitude.

In this study, we construct the new solitary wave solutions for two well known nonlinear wave equations whose describe the wave propagation in nonlinear media. We implemented a new technique which is extension of modified rational expansion method on nonlinear three-dim modified Kortewege–de Vries Zakharov–Kuznetsov and extended Zakharov–Kuznetsov equations. The new results are elliptic, hyperbolic, trigonometric, rational type which are representing to solitary waves, periodic waves, traveling waves, kink-antikink solitons, dark-bright solitons. The physical behavior of some calculated results represented graphical by using the software Mathematica. The determined results are new, more general and have numerous applications in the area of nonlinear sciences including nonlinear plasma, fiber optics, mechanics, quantum physics, laser physics, fluid physics, engineering and other types of physical sciences. The demonstrated results show that new technique is efficient, fruitful, powerful and reliable to study analytically different types of higher order complex NLPDEs arises in various types of applied sciences.

Progress in the development of new tools for modem apphed mathematics resulted in a better scientific under­ standing of nonlinear waves in various fields of nonlinear optics, hydrodynamics, and plasmas. The universahty of the equations describing wave processes is one of the most important components of this success. Nonlinear Schrodinger equation (NLS) and Korteweg de Vries equation are among best illustrations of successful application of apphed mathematics as a tool to analyze various nonlinear phenomena ranging from optical communications and Bose con­ densation to ocean waves. Solutions of nonlinear equations usually result in the formation of singularities , coherent structures or solitary waves. Examples of the corresponding phenomena can be observed in filamentation of laser beams in nonlinear media, wave breaking in hydrodynamics, collapse, and Langmuir waves in plasmas. NLS in dimension one (ID) is a basis for the description of modem optical fiber communication systems. A key technological tool for development of ultrafast high-bitrate optical communication lines is a dispersion management. A dispersion-managed optical fiber is described by NLS with periodic variation of dispersion along an optical line which dramatically reduces pulse broadening. Solitary wave solutions in such systems are important information carriers in optical lines. NLS in 2D describes a stationary self-focusing of light in optical media with Kerr nonlinearity, Bose-Einstein condensate in plane geometry as well as a self-focusing in plasma. Due to self-focusing the amplitude of an NLS solution tends to infinity after finite distance of propagation, representing a blow up phenomenon. The blow up is accompanied by a dramatic contraction of the spatial size of solution, which is called collapse. Near the collapse point NLS loses its applicability and there is usually a quahtative change in the underlying nonlinear phenomena, the new mechanisms become important such as dissipation, inelastic two- and three-body collisions, which can cause a loss of atoms from the Bose-Einstein condensate, breakdown of slow envelope approximation in nonlinear optical media and plasma density depletion. If optical power is very large in optics or if a number of particles in the BoseEinstein condensate significantly exceeds the critical number, then collapse turbulence occurs with random positions of collapses in space and time. A study of collapse regularization is important in describing collapse turbulence and to be able to go beyond an individual collapse event. 3D NLS is a model for non-stationary self-focusing of hght in optical media with Kerr nonlinearity, Bose-Einstein condensate in 3D geometry and laser-plasma interactions. Self-focusing results in collapse, which is of a different type as compare with 2D, and it is called weak collapse. NLS in 3D is a model for ultrashort pulse (optical bullet) that is a key for current developments in microfabrication of photonics devices by means of intense femtosecond laser pulses. Regularization of collapse for such pulses requires an extension of the model beyond NLS and a study of the full set of MaxwellSs equations. Maxwell's equations are also necessary to study ultrashort hght propagation through thin films and guided-wave metamaterials. New challenges for nonlinear optics and apphed mathematics include description of nonlinear optical phenomena in negative-refractive-index metamaterials, which are artificial nano-composites materials that show great promise for opening the new avenues in manipulating light. Light propagating through them could exhibit extraordinary properties and brings a wealth of novel phenomena to nonlinear optics. Electrodynamical properties of metamaterials result from many factors: plasmonic oscillations in metallic nanostructures, multi-scale interaction of near field electromagnetic resonances with optical far field dynamics, surface roughness of nanostructures, high values of local fields that cause nonlinear response and etc. It has recently been shown that mathematical modeling of hght interaction with thin film metamaterials requires the use of classical hydrodynamics models. Therefore, metamaterials represent a new research

The generalized Kuramoto-Sivashinsky equation is considered. The Bogning-Djeumen Tchaho-Kofane method is used to study approximate solitary wave solutions of this equation. It is shown that the generalized Kuramoto-Sivashinsky equation has hybrid solitary wave solutions which are a combination of bright, kink, and dark solitary wave profiles, respectively. The possibility to alter the amplitude of each individual solitary wave of the combination allows to choose which profiles are dominant. Numerical simulations corroborate the analytical predictions with a good accuracy. Further numerical simulations reveal that the hybrid solitary wave solutions taken as a perturbation of the trivial solution remain stable for a relatively long time, hence might be observed in experiments.

Ginzburg–Landau equations describe a wide range of phenomena involving superconductivity, superfluidity, etc. In the present paper, Ginzburg–Landau equations involving distinct laws are considered, and as a consequence, their solitary waves in the presence of perturbation terms are formally derived using the Kudryashov method. The effect of Kerr and parabolic laws on the dynamics of solitary waves is examined in detail. The outcomes of the current paper present suitable ways to control the width and amplitude of solitary waves. The authors believe that the results reported in the current study will contribute significantly to studies related to Ginzburg–Landau equations with distinct laws.

A variational approach was used to derive a set of Serre equations for fully nonlinear, dispersive waves in channels of arbitrary cross section. A family of travelling waves was found, as well as the relation between amplitude and celerity of solitary waves. An upper bound is proposed for the solitary wave amplitude as a function of the Froude number in trapezoidal cross-sectional canals, and it showed good agreement with existing theory. For waves of moderate amplitude, cnoidal waves result with a soliton limit; these waves and their properties (celerity and wave number) are written as functions of the channel bank slope and channel bank curvature. The theoretical findings are in agreement with well-established results in the literature, in particular with more-recent Boussinesq-type theories. A validation is proposed against existing experimental data.

This study aims to examine the properties of the nonplanar (cylindrical and spherical) ion-acoustic solitary waves (SWs) and cnoidal waves (CWs) in a collisionless, unmagnetized electron-ion (EI) plasma having a Cairns–Tsallis distribution for the electrons. This study is structured around two main lines. The first trend involves deriving the nonplanar Korteweg-de Vries (KdV) equation by utilizing the method of reductive perturbation (MRP). This equation describes small-amplitude (non)planar acoustic waves (AWs). Furthermore, the nonplanar Kawahara equation (KE) is formulated to examine the significant magnitude of planar and nonplanar SWs and CWs. The current plasma model supports compressive and rarefactive IA solitary and cnoidal structures, depending upon the associated physical factors such as the nonextensive parameter (nonextensivity) and nonthermal parameter (nonthermality). When the plasma compositions reach some critical values, such as the critical value of nonthermality, the coefficient of the quadratic nonlinear term vanishes. Hence, both nonplanar modified KdV (mKdV) equation and modified KE (mKE) with cubic nonlinearity are derived to accurately depict the dynamics of both small and large amplitudes of nonplanar SWs and CWs and any other structures related to this family of evolution equations. The influences of the nonextensivity and nonthermality on the profile of (non)planar KdV soliton and the (non)planar Kawahara SWs and CWs are numerically examined using some semi-analytical and numerical approximations. Also, the impact of the nonextensivity on the profile of (non)planar mKdV soliton and the (non)planar modified Kawahara SWs and CWs is reported. It is tracked down that the variety of different plasma parameters significantly alters the characteristic properties of the small and large amplitude ion-acoustic waves (IAWs) discussed by the nonplanar KdV-type equations.

This study numerically investigates the nonlinear interaction of head-on solitary waves in a granular chain (a nonintegrable system) and compares the simulation results with the theoretical results in fluid (an integrable system). Three stages (the pre-in-phase traveling stage, the central-collision stage, and the post-in-phase traveling stage) are identified to describe the nonlinear interaction processes in the granular chain. The nonlinear scattering effect occurs in the central-collision stage, which decreases the amplitude of the incident solitary waves. Compared with the leading-time phase in the incident and separation collision processes, the lagging-time phase in the separation collision process is smaller. This asymmetrical nonlinear collision results in an occurrence of leading phase shifts of time and space in the post-in-phase traveling stage. We next find that the solitary wave amplitude does not influence the immediate space-phase shift in the granular chain. The space-phase shift of the post-in-phase traveling stage is only determined by the measurement position rather than the wave amplitude. The results are reversed in the fluid. An increase in solitary wave amplitude leads to decreased attachment, detachment, and residence times for granular chains and fluid. For the immediate time-phase shift, leading and lagging phenomena appear in the granular chain and the fluid, respectively. These results offer new knowledge for designing mechanical metamaterials and energy-mitigating systems.

A Korteweg-de Vries (KdV) equation with a linear Landau damping term describing weakly nonlinear and weakly dispersive ion-acoustic waves in an electron-positron-ion plasma is derived. It is found that the Landau damping causes the solitary wave amplitude to decay with time. It is also found that in absence of Landau damping, both linear wave phase velocity and solitary wave amplitude decrease with the increase of positron density, whereas, both increase with the increase of positron temperature. On the other hand, the Landau damping rate decreases with the increase of both positron density and temperature.

By using the reductive perturbation method (RPM), a nonlinear Zakharov-Kuznetsov (ZK) equation for ion-acoustic solitary waves (IASWs) is derived for a magnetized plasma in which the electrons are nonextensively distributed. The combined effects of electron nonextensivity, strength of magnetic field, and obliqueness on the ion acoustic (IA) solitary profile are analyzed. Three different ranges of the nonextensive q-parameter are considered. It is observed that the system may support both compressive as well as rarefactive solitons. The magnetic field has no effect on the amplitude of solitary waves whereas the obliqueness affects both the amplitude as well as the width of the solitary wave structures.

The effects of nonextensive electrons on nonlinear ion acoustic waves in dusty negative ion plasmas with ion-dust collisions are investigated. Analytical results show that both solitary and shock waves are supported in this system. The wave propagation is governed by a Korteweg-de Vries Burgers-type equation. The coefficients of this equation are modified by the nonextensive parameter q. Numerical calculations indicate that the amplitude of solitary wave and oscillatory shock can be obviously modified by the nonextensive electrons, but the monotonic shock is little affected.