Nonlinear Wave Systems Research Articles

Turbulence in Integrable Systems

Nonlinear wave systems integrable by Inverse Scattering Method (ISM) could demonstrate a complex behavior that demands the statistical description. The theory of this description composes a new chapter in the theory of wave turbulence—Turbulence in Integrable Systems. All systems integrable by ISM are separated in two classes: strongly and weakly integrable. Systems of both classes have infinite array of motion constants but only for strongly integrable systems this array is complete. As a result, the scattering is trivial in these systems. It means that all the collision terms in kinetic equations of arbitrary high order are identically zero. The examples of strongly integrable systems are: KdV, NLSE, and KP‐2 equations. In strongly integrable systems one can choose as initial data a statistically homogenous random field with a given pair correlation function such that this function is invariant in time. The spatial spectrum of an equilibrium state can be chosen in an arbitrary way. In weakly integrable systems (KP‐1, three‐wave system, etc) the kinetic equations are nontrivial. They have infinite but incomplete set of motion constants. These kinetic equations have infinite amount of Rayley–Jeans‐type stationary solutions, though their general stationary solutions are not explored yet.

Localized Waves in Nonlinear Systems with Spatially Inhomogeneous Nonlinearity

A type of (2+1)-dimensional nonlinear Schrödinger equation with spatially inhomogeneous nonlinearity and an external potential is studied. It is found that special external potentials and spatially nonlinearities can support nonlinear localized waves.

Ab initio self-consistent laser theory and random lasers

We review our recent work leading to steady-state solutions of the semiclassical (Maxwell–Bloch) equations of a laser. These are coupled nonlinear partial differential equations in space and time which have previously been solved either by fully time-dependent numerical simulations or by using major approximations which neglect nonlinear modal interactions and/or the openness of the laser system. We have found a time-independent technique for determining these stationary solutions which can treat lasers of arbitrary complexity and degree of openness. Our method has been shown to agree with time-dependent numerical solutions to high accuracy and has been applied to find the electric field patterns (lasing modes) of random lasers, which lack a laser cavity and are so strongly damped that the linear system has no detectable resonances. Our work provides a link between an important nonlinear wave system and the field of quantum/wave chaos in linear systems.

Applications of HTA and EHTA to YTSF Equation

New approaches, homoclinic test approach (HTA) and extended homoclinic test approach (EHTA), are proposed to seek solitary-wave solution of high dimensional nonlinear wave system. Exact periodic solitary-wave, periodic soliton, cross solitary-wave and doubly periodic wave solutions for YTSF equation are obtained using HTA and EHTA, respectively.

Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.

Damping of Periodic Waves in Physically Significant Wave Systems

Damping of periodic waves in the classically important nonlinear wave systems—nonlinear Schrödinger, Korteweg–deVries (KdV), and modified KdV—is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus,m=m(t). The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution whenm= 0to soliton waves whenm= 1.

WEAK REGULAR REFLECTION FOR THE NONLINEAR WAVE SYSTEM

We prove local existence of a solution to a Riemann problem for the two-dimensional nonlinear wave system using the approach by Čanić, Keyfitz, Kim and Lieberman. We consider initial data resulting in weak regular shock reflection. By writing the problem in self-similar coordinates, we obtain a mixed type system and a free boundary problem for the subsonic flow and the position of the reflected shock. We reformulate this problem using a second order equation for density with mixed boundary conditions and an ordinary differential equation describing the location of the reflected shock. The main difficulty in the study of this second order problem for density is that the operator is degenerate elliptic along the sonic circle. We regularize and modify the operator, and we show existence of solutions to the regularized problems. We prove that the sequence of solutions to the regularized problems is precompact, and, using uniform local ellipticity, covering and diagonalization techniques, we obtain a local solution to the original problem.

Observation of acoustic turbulence in a system of nonlinear second sound waves in superfluid He4

The results of recent studies of acoustic turbulence in a system of nonlinear second sound waves in a high-quality resonator filled with superfluid He4 are discussed. It is found that when the driving amplitude is increased sufficiently, a steady-state direct wave cascade, involving a flux of energy towards high frequencies, is formed. The wave amplitude distribution follows a power law over a wide range of frequencies. Development of a decay instability at high driving amplitudes results in the formation of subharmonics of the driving frequency, and to a backflow of energy towards the low-frequency spectral domain, in addition to the direct cascade.

Embedded Soliton Solutions: A Variational Study

Embedded solitons(ES) represent solitary waves which reside inside the continuous spectrum of a nonlinear-wave system. This type of solitons was first reported by Yang et al. [1] in optical models characterized by opposing dispersion and competing nonlinearities at fundamental and second harmonics. More specifically, optical media with quadratic χ and cubic χ nonlinear susceptibilities can support ES solutions. The evolution of ES is governed by the coupled nonlinear partial differential equations [2]:

Turbulence of Second Sound Waves in Superfluid 4He: Effect of Low-Frequency Resonant Perturbations

We report the results of investigations of acoustic turbulence in a system of nonlinear second sound waves in a high-quality resonator filled with superfluid 4He. It was observed that subharmonics of a periodic driving force applied to the system may be generated via a parametric instability. We find that application of an additional low-frequency pumping to the turbulent system results in the generation of waves at combination frequencies of the driving forces and also leads to substantial changes in the energy spectrum of the acoustic oscillations.

Persistence of travelling waves for a coupled nonlinear wave system

In this paper, we study a sort of coupled nonlinear wave system. Special attention is paid to the question of the existence of heteroclinic orbits of the associated ordinary differential equation from the geometric singular perturbation point of view. We prove that a solitary wave persists when the perturbation parameter is suitably small. This argument does not require an explicit expression for the original coupled nonlinear wave system.

The Triple Point Paradox for the Nonlinear Wave System

We present numerical solutions of a two-dimensional Riemann problem for the non- linear wave system which is used to describe the Mach reflection of weak shock waves. Robust low order as well as high resolution finite volume schemes are employed to solve this equation formulated in self-similar variables. These, together with extreme local grid refinement, are used to resolve the solution in the neighborhood of an apparent but mathematically inadmissible shock triple point. Rather than observing three shocks meeting in a single standard triple point, we are able to detect a primary triple point containing an additional wave, a centered expansion fan, together with a se- quence of secondary triple points and tiny supersonic patches embedded within the subsonic region directly behind the Mach stem. An expansion fan originates at each triple point. It is our opinion that the structure observed here resolves the von Neumann triple point paradox for the nonlinear wave system. These solutions closely resemble the solutions obtained in (A. M. Tesdall and J. K. Hunter, SIAM J. Appl. Math., 63 (2002), pp. 42-61) for the unsteady transonic small disturbance equation.

One-dimensional and two-dimensional arrays of nanoholes generated by laser in the semiconfined configuration

We have shown that nanoporosity can be generated on metal surfaces by nanosecond laser-matter interactions in the semiconfined configuration. The scanning electron microscope analysis has shown that nanoholes of ∼25–50nm in diameter, arranged in one-dimensional (1D) and two-dimensional (2D) irregular and regular arrays, have been formed. The interpretation is based on the generation of a dispersive, dissipative system of nonlinear solitary plasma waves (humps) that leave temperature/pressure fingerprints on the metal surface. It has been shown that the 1D irregular array of nanoholes can be interpreted as a result of the irregular string of solitary humps obtained by numerical simulation based on the Benney pd equation with the Gaussian perturbation. The 2D random array of nanoholes can be interpreted as a result of random solitary humps that can be obtained by numerical simulation from the Benney equation with the periodic perturbation. The regular string of nanoholes has been shown to appear as a result of breather modes (bound state of solitons), the numerical simulation of which has been based on the Boussinesq equation. The regular 2D array of nanoholes has been interpreted as fingerprints of breather modes, in agreement with the result of the numerical simulation of Tajiri and Murakami, [J. Math. Phys. 34, 2400 (1993)], based on the Kadomtsev-Petviashvili equation.

LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS

We study nonlinear dispersive wave systems described by hyperbolic PDE's in ℝdand difference equations on the lattice ℤd. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

TRANSONIC REGULAR REFLECTION FOR THE NONLINEAR WAVE SYSTEM

We consider Riemann data for the nonlinear wave system which result in a regular reflection with a subsonic state behind the reflected shock. The problem in self-similar coordinates leads to a system of mixed type and a free boundary value problem for the reflected shock and the solution in the subsonic region. We show existence of a solution in a neighborhood of the reflection point.

New exact solution structures and nonlinear dispersion in the coupled nonlinear wave systems

Abstract In this Letter, to further understand the role of nonlinear dispersion in coupled nonlinear wave systems in both real and complex fields, we study the coupled Klein–Gordon equations with nonlinear dispersion in real field (called CKG ( m , n , k ) equation) and ( 2 + 1 ) -dimensional generalization of coupled nonlinear Schrodinger equation with nonlinear dispersion in complex field (called GCNLS ( m , n , k ) equation) via some transformations. As a consequence, some types of solutions are obtained, which contain compactons, solitary pattern solutions, envelope compacton solutions, envelope solitary pattern solutions, solitary wave solutions and rational solutions.

Weak Turbulence for a Vibrating Plate: Can One Hear a Kolmogorov Spectrum?

We study the long-time evolution of waves of a thin elastic plate in the limit of small deformation so that modes of oscillations interact weakly. According to the theory of weak turbulence (successfully applied in the past to plasma, optics, and hydrodynamic waves), this nonlinear wave system evolves at long times with a slow transfer of energy from one mode to another. We derive a kinetic equation for the spectral transfer in terms of the second order moment. We show that such a theory describes the approach to an equilibrium wave spectrum and represents also an energy cascade, often called the Kolmogorov-Zakharov spectrum. We perform numerical simulations that confirm this scenario.

Equilibrium theory of coherent vortex and zonal jet formation in a system of nonlinear Rossby waves

The problem of coherent vortex and zonal jet formation in a system of oceanic or planetary nonlinear Rossby waves is considered from the point of view of the late time steady state achieved by free decay of a given initial state. Statistical equilibrium equations respecting all conservation laws are constructed for a broad class of models, generalizing those derived previously for two-dimensional inviscid Euler flow. Jetlike solutions are ubiquitous, with large coherent vortices existing only when there is a background flow whose velocity locally cancels the beta effect.

Nonequilibrated Oscillations of Coherence in Coupled Nonlinear Wave Systems

We show that a conservative system of a pair of coupled incoherent nonlinear waves exhibits huge oscillations of coherence, which are characterized by a recurrent transfer of noise fluctuations between the coupled waves. This sustained oscillatory behavior is in contradiction with the expected irreversible evolution towards equilibrium. As a consequence, the process of coherence transfer is characterized by a reduction of nonequilibrium entropy, which violates the H theorem of entropy growth inherent to the kinetic theory.

Free Boundary Problems for Nonlinear Wave Systems: Mach Stems for Interacting Shocks

We study a family of two-dimensional Riemann problems for compressible flow modeled by the nonlinear wave system. The initial constant states are separated by two jump discontinuities, $x = \pm \kappa_a y$, which develop into two interacting shock waves. We consider shock angles in a range where regular reflection is not possible. The solution is symmetric about the y-axis and on each side of the y-axis consists of an incident shock, a reflected compression wave, and a Mach stem. This has a clear analogy with the problem of shock reflection by a ramp. It is well known that no triple point structure exists in which incident, reflected, and Mach stem shocks meet at a point. In this paper, we model the reflected wave by a continuous function with a singularity in the derivative. This fails to be a weak solution across the sonic line. We show that a solution to the free boundary problem for the Mach stem exists, and we conjecture that the global solution can be completed by the construction of a reflected shock, by a similar free boundary technique. The point of our paper is the capability to deal analytically with a Mach stem by solving a free boundary problem. The difficulties associated with the analysis of solutions containing Mach stems include (1) loss of obliqueness in the derivative boundary condition corresponding to the jump conditions across the Mach stem, and (2) loss of ellipticity at the formation point of the Mach stem. We use barrier functions to show that for sufficiently large values of $\kappa_a$ the subsonic solution is continuous up to the sonic line at the Mach stem.