Statistical Approach to Modulational Instability in Nonlinear Discrete Systems

Bifurcations of travelling wave solutions in the discrete NLS equations

We overview theoretical and experimental results on spatial optical solitons excited in arrays of nonlinear waveguides. First, we briefly summarize the basic properties of the discrete nonlinear Schrödinger (NLS) equation frequently employed to study spa

In this paper, the Nonlocal Discrete Nonlinear Schrodinger (DNLS) equation that interpolates the Nonlocal Ablowitz-Ladik DNLS and the Nonlocal Cubic DNLS equations and its stability are studied in detail. The solution of the Nonlocal SNLD equation is a soliton wave in the form of a Gaussian ansatz obtained using the method of Variational Approximation (VA). The stability of the solution is also analyzed using the VA. These semi-analytical results are then compared to numerical results. The soliton and its stability obtained via VA is concluded to be having a fairly good conformity with numerical results.

Exact anharmonic-localized-mode solutions to the discrete nonlinear Schrodinger (NLS) equation defined in a d -dimensional version of the simple cubic lattice are obtained in terms of lattice Green's functions described by the ordinary and modified Bessel functions. A stationary or nonmoving localized mode is shown to be stable, while a moving one disperses, the rate of dispersing being smaller for a smaller moving velocity. For \(d{\geqslant}3\) there exists a critical value of the lattice nonlinearity for the appearance of such a localized mode, and its spatial localization becomes pronounced as d increases. For d =1 the solution reduces in the continuum limit to the conventional one-soliton solution of the NLS equation. A very brief study is also made on the two-localized-mode problem.

The persistence of stationary and travelling single-humped localized solutions in the spatial discretizations of the nonlinear Schrödinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is considered. Constraints on the nonlinear function are found from the condition that the second-order difference equation for stationary solutions can be reduced to the first-order difference map. The discrete NLS equation with such an exceptional nonlinear function is shown to have a conserved momentum but admits no standard Hamiltonian structure. It is proved that the reduction to the first-order difference map gives a sufficient condition for existence of translationally invariant single-humped stationary solutions. Another constraint on the nonlinear function is found from the condition that the differential advance–delay equation for travelling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction gives a necessary condition for existence of single-humped travelling solutions. The nonlinear function which admits both reductions defines a four-parameter family of discrete NLS equations which generalizes the integrable Ablowitz–Ladik lattice. Particular travelling solutions of this family of discrete NLS equations are written explicitly.

The one-dimensional ferromagnet XXZ spin chain with Dzyaloshinsky–Moriya (DM) interaction, recently introduced by Djoufack and coworkers is reexplored, with a particular attention carried on their found discrete nonlinear Schrödinger (DNLS) equation, governing the quantum breathers states behaviors. This DNLS equation admits exact bright compacton and peakon-like solutions, where analytical expressions, the existence and stability criteria are found and used to obtain their quantized energy levels. Using the semi-discrete multiple-scale method, the DNLS equation is reduced to the extended nonlinear Schrödinger equation which consists of the basic NLS equation with additional nonlinear dispersive terms Via the bifurcation diagram and Liapunov exponents, we provide a summary of essential dynamics and show that the equation would admit several forms of solutions among which the localized Hartree n compacton and peakon-like boson quantum breathers states. Furthermore, we notice that on the contrary to the classical outcomes where amplitudes of both solutions are free parameters, the amplitudes for quantum states are not free since the obtained solutions need to be normalized. The stationary localization of both compacton and peakon-like states is confirmed by numerical simulations performed on the DNLS equation.

The field experiment is conducted from April 16, 2005 to July 20, 2005 at Wenchang area east of Hainan Island (19°35'N, 112°E) of China. Internal wave packets are observed frequently with thermistor chains during the experiment. Meanwhile, internal waves are also detected from a synthetic aperture radar (SAR) image on June 19, 2005 and several other moderate-resolution imaging spectroradiometer (MODIS) images near a mooring position. The distance between the positive and negative peaks induced by the internal wave can be obtained from satellite images. Combined with remote sensing images and in situ data, a new method to inverse the amplitude of the internal wave is proposed based on a corrected nonlinear Schrodinger (NLS) equation. Two relationships are given between the peak-to-peak distance and the characteristic wavelength of the internal wave for different nonlinear and dispersion coefficients. Based on the satellite images, the amplitude inversion of the internal waves are carried out with the NLS equation as well as the KdV equation. The calculated amplitudes of the NLS equation are close to the observation amplitude which promise the NLS equation a reliable method.

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrodinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, \({S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}\), we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, \({I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}\), and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

We investigate the procedures of discretization of the integrable nonlinear Schrodinger dynamical system, well known as the Ablowitz–Ladik equation, the corresponding symplectic structures, and the finite-dimensional invariant reductions. We develop an efficient scheme of invariant reduction of the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter. We construct a finite set of recurrence algebraic regular relations that allows one to generate solutions of the discrete nonlinear Schrodinger dynamical system and discuss the related functional spaces of solutions. Finally, we analyze the Fourier-transform approach to the study of the set of solutions of the discrete nonlinear Schrodinger dynamical system and its functional-analytic aspects.

We present a method by which one-dimensional nonlinear soliton and kink Schrodinger equations can be solved in closed form. The hermitean nonlinear soliton operator may contain up to second derivatives of the wave function and the vanishing condition must hold. The method is applied to solve known nonlinear Schrodinger equations for one-soliton and one-kink solutions and, by inverting the procedure, to derive new operators with wave packet solutions of algebraic and arbitrary shapes. One of them is equivalent to the Derivative Nonlinear Schrodinger equation.

The spatio-temporal behaviors of possible solutions of driven and damped nonlinear Schrodinger and sine-Gordon equations are analyzed. In the first part, the nonlinear Schrodinger equation is considered also as the limiting case corresponding to weak driving and damping in the sine-Gordon equation, with driving frequency close to one. It is shown that in this limit a phase-locked breather with m/n = 1/1 exists and undergoes period-doubling bifurcations. The close analogy to the dynamics of a phase-locked Schrodinger soliton is demonstrated in detail. In the second part, for smaller driving frequencies, the existence and stability of subharmonically phase-locked breathers is discussed. The quasi-periodic route to chaos is re-investigated and special attention is given to the length-dependence of the results. The findings are compared with predictions of simpler models in nonlinear dynamics. The whole paper consists of analytical and numerical results. For the latter, improved diagnostic tools have been developed.

The integrable vector nonlinear Schrodinger (vector NLS) equation is investigated by using Zakharov-Shabat (ZS) scheme. We get a Lax pair formulation of the vector NLS model. Multi-soliton solution of the equation is also derived by using inverse scattering method of ZS scheme. We also find that there is an elastic and inelastic collision of the bright and dark multi-solitons of the system.

The NLS(n, n) equation: Multi-hump compactons and their stability and interaction scenarios

Based on the unified theory by the present authors (Kanagawa et al., J. Fluid Sci. Tech., 5, 2010), the Korteweg-de Vries-Burgers (KdVB) equation and the nonlinear Schrodinger (NLS) equation with an attenuation term for weakly nonlinear waves in bubbly liquids are re-derived from a system of bubble-liquid mixture model equations composed of the conservation equations of mass and momentum, the Keller equation for bubble dynamics, and supplementary equations. We show that the re-derived KdVB equation and NLS equation are essentially the same as those derived from a system of two-fluid model equations except for the coefficients of nonlinear, dissipation, and dispersion terms. The differences in these coefficients are studied in detail, and we find that for the case of KdVB equation, the mixture model is valid only for sufficiently small initial void fractions. On the other hand, for the case of NLS equation, the range of validity of the mixture model depends on not only the initial void fraction but also the wavenumber concerned.

The modulational instability in the class of NLS equations is discussed using a statistical approach. A kinetic equation for the two-point correlation function is studied in a linear approximation, and an integral stability equation is found. The modulational instability is associated with a positive imaginary part of the frequency. The integral equation is solved for different types of initial distributions (delta-function, Lorentzian) and the results are compared with those obtained using a deterministic approach. The differences between modulation instability of the normal NLS equation and derivative NLS equations is emphasized.