Two-dimensional nonlinear Schrödinger equation with random radial data

The two-dimensional cubic nonlinear Schrodinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. Coupled two-dimensional cubic nonlinear Schrodinger equations are used to describe the interaction of electromagnetic waves with different polarizations in nonlinear optics. In this chapter, we solve these equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of this type are obtained, many of which are periodic, quasi-periodic, aperiodic, and singular solutions that may have practical significance. The Davey–Stewartson equations are used to describe the long time evolution of three-dimensional packets of surface waves. Assuming that the argument functions are quadratic in spatial variables, we find various exact solutions for the Davey–Stewartson equations.

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrodinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grunwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrodinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

The (1+1)-dimensional theory of envelope soliton propagations rests on the nonlinear Schrodinger (NLS) equation, which reads: $$ iAu_z + Bu_{xx} + Cu|u|^2 = 0 $$ (1) x is the pulse shape variable, either transverse or longitudinal (time variable t). A well- known, but essential property of the NLS equation is to be completely integrable by means of the inverse scattering transform (IST) method for any value of the real coefficients A, B, and C. The most simple generalization of the NLS equation to (2+1) dimensions is the so-called two-dimensional nonlinear Schrodinger equation (2D NLS), that reads: $$ iAu_z + Bu_{xx} + Cu_{yy} + Du|u|^2 = 0 $$ (2) Equation (2) is not integrable, whatever the value of the coefficients is, and does not admit any stable localized solution.

Nonlinear Schrodinger equation (NLSE) is now one of the prominent of modern physics, mathematics and chemistry. Over these fields, the NLSE is also applied in new emerging fields such as quantum information and econophysics. In this paper we investigate for new exact solutions of higher order nonlinear Schrodinger’s equation. This method allows to carry out the solution process of nonlinear wave equations more thoroughly and conveniently by computer algebra systems such as the Maple and Mathematica. In addition to providing a different way of solving the Schrodinger equation for such systems, the simplicity of the algorithm renders it a great pedagogical value.

We use a statistical approach to investigate the modulational instability (Benjamin-Feir instability) in several nonlinear discrete systems: the discrete nonlinear Schrodinger (NLS) equation, the Ablowitz-Ladik equation, and the discrete deformable NLS equation. We derive a kinetic equation for the two-point correlation function and use a Wigner-Moyal transformation to write it in a mixed space-wave-number representation. We perform a linear stability analysis of the resulting equation and discuss the obtained integral stability condition using several forms of the initial unperturbed spectrum (Lorentzian and δ-spectrum). We compare the results with the continuum limit (the NLS equation) and with previous results.

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.

Presented is a set of integrable multi-component hybrid nonlinear Schrodinger (MCHNS) equations. Each multi-component equation is a superposition of the nonlinear Schrodinger (NS) equation and the derivative nonlinear Schrodinger (DNS) equation. For the MCHNS equations, the inverse scattering formulation is given. The gauge transformation relating 2-component hybrid nonlinear Schrodinger equation with the Manakov equation is explicitly shown. This also confirms that the former is integrable since the latter is integrable.

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

Nonlinear Schrodinger equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant when coupling quantum models to classical models. With the aim of describing the semiclassical limit of the three-dimensional (3D) Schrodinger--Poisson system with an additional harmonic potential, we study some semiclassical limits of the Hartree equation with harmonic potential in space dimension $n \geq 2$. The harmonic potential is confining and causes focusing periodically in time. We prove asymptotics in several cases, showing different possible nonlinear phenomena according to the interplay of the size of the initial data and the power of the Hartree potential. In the case of the 3D Schrodinger--Poisson system with harmonic potential, we can give only a formal computation since the need for modified scattering operators for this long-range scattering case goes beyo...

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.

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.

The optical soliton solution of nonlinear complex models holds significant importance in nonlinear optics and communication systems. Considering nonlinear complex models, often described by equations like the nonlinear Schrodinger equation (NLSE), plays a crucial role in defining the balance between dispersive and nonlinear effects, enabling the formation and maintenance of solitons over long distances. This stability is crucial for signal integrity in optical communication systems. The investigation of optical soliton solutions from nonlinear complex models is sometimes complicated. With this in mind, we employed an effective method of extended Tanh function with a Riccatti differential equation to retrieve the dark, singular and periodic wave solutions for the weakly nonlocal nonlinear Schrodinger equation with parabolic law. The obtained solutions were verified by back-substitution in the original equations, with the aid of a Mathematica to affirm the robustness of the chosen approach. Respective 2D and 3D graphs for some of the obtained results was portrayed by choosing suitable values of the parameters that were involved. An analysis of instability that results in the modulation of the steady-state as a result of co-action between the nonlinear and dispersive effects was performed on the proposed model where the condition for stable wave under small perturbation was obtained and presented. The gain spectrum plot for the modulation instability was portrayed.