Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

Stability Of Solitary Wave Solutions For Equations Of Short And Long Dispersive Waves

In this paper, we consider the existence and stability of a novel set of solitary-wave solutions for two models of short and long dispersive waves in a two layer fluid. We prove the existence of solitary waves via the Concentration Compactness Method. We then introduce the sets of solitary waves obtained through our analysis for each model and we show that them are stable provided the associated action is strictly convex. We also establish the existence of intervals of convexity for each associated action. Our analysis does not depend of spectral conditions. © 2006 Texas State University.2006118Albert, J., Angulo, J., Existence and stability of ground-state solutions of a Schrödinger-KdV system (2003) Proc. Roy. Soc. Edinburgh, Sect. A, 133 (5), pp. 987-1029Angulo, J., Montenegro, J.F., Existence and evenness of solitary-wave solutions for an equation of short and long dispersive waves (2000) Nonlinearity, 13 (5), pp. 1595-1611Bekiranov, D., Ogawa, T., Ponce, G., Interaction equation for short and long dispersive waves (1998) J. Funct. Anal., 158, pp. 357-388Bekiranov, D., Ogawa, T., Ponce, G., Weak solvability and well-posedness of a coupled Schrödinger- Korteweg de Vries equation for capillary-gravity wave interactions (1997) Proc. Amer. Math. Soc., 125 (10), pp. 2907-2919Cazenave, T., Lions, P.-L., Orbital stability of standing waves for some nonlinear Schrödinger equations (1982) Comm. Math. Phys., 85, pp. 549-561Fernandez, A., Linares, F., (2004) Well-posedness for the Schrödinger-korteweg-de Vries Equation, , PreprintFunakoshi, M., Oikawa, M., The resonant interaction between a long internal gravity wave and a surface gravity wave packet (1983) J. Phys. Soc. Japan, 52, pp. 1982-1995Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry II (1990) J. Funct. Anal., 94, pp. 308-348Kawahara, T., Sugimoto, N., Kakutani, T., Nonlinear interaction between short and long capillary-gravity waves (1975) J. Phys. Soc. Japan, 39, pp. 11379-11386Levandosky, S., Stability and instability of fourth-order solitary waves (1998) J. Dynam. Diff. Eqs., 10, pp. 151-188Lin, C., Orbital stability of solitary waves of the nonlinear Schrödinger-KDV equation (1999) J. Partial Diff. Eqs., 12, pp. 11-25Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case, part 1 (1984) Ann. Inst. H. Poincaré, Anal. Non Linéare, 1, pp. 109-145Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case, part 2 (1984) Ann. Inst. H. Poincaré, Anal. Non Lineare, 4, pp. 223-283Lopes, O., Variational systems defined by improper integrals (1998) International Conference on Differential Equations, pp. 137-151. , Magalhaes, L. et al, World ScientificLopes, O., Nonlocal Variational Problems Arising in Long Wave Propagation, , to appearShatah, J., Stable standing waves of nonlinear Klein Gordon equations (1983) Commun. Math. Phys., 91, pp. 313-327Tsutsumi, M., Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation (1993) Math. Sci. Appl., 2, pp. 513-52

Exact Solutions for Domain Walls in Coupled Complex Ginzburg - Landau Equations

The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the evolution of slowly varying wave packets in nonlinear dissipative media. A front (shock) is a transient layer between a plane-wave state and a zero background. We report exact solutions for domain walls, i.e., pairs of fronts with opposite polarities, in a system of two coupled CGLEs, which describe transient layers between semi-infinite domains occupied by each component in the absence of the other one. For this purpose, a modified Hirota bilinear operator, first proposed by Bekki and Nozaki, is employed. A novel factorization procedure is applied to reduce the intermediate calculations considerably. The ensuing system of equations for the amplitudes and frequencies is solved by means of computer-assisted algebra. Exact solutions for mutually-locked front pairs of opposite polarities, with one or several free parameters, are thus generated. The signs of the cubic gain/loss, linear amplification/attenuation, and velocity of the coupled-front complex can be adjusted in a variety of configurations. Numerical simulations are performed to study the stability properties of such fronts.Comment: Journal of the Physical Society of Japan, in pres

Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States

The system of a cold atomic gas in an optical lattice is governed by two factors: nonlinearity originating from the interparticle interaction, and the periodicity of the system set by the lattice. The high level of controllability associated with such an arrangement allows for the study of the competition and interplay between these two, and gives rise to a whole range of interesting and rich nonlinear effects. This review covers the basic idea and overview of such nonlinear phenomena, especially those corresponding to extended states. This includes "swallowtail" loop structures of the energy band, Bloch states with multiple periodicity, and those in "nonlinear lattices", i.e., systems with the nonlinear interaction term itself being a periodic function in space.Comment: 39 pages, 21 figures; review article to be published in a Special Issue of Entropy on "Non-Linear Lattice