Erratum: Sakhabutdinov et al. Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber. Fibers 2021, 9, 1

The authors wish to make a change to the author names (adding a new author—Ildaris M [...]

Dynamical Behavior and the Classification of Single Traveling Wave Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

In this paper, the dynamical properties and the classification of single traveling wave solutions of the coupled nonlinear Schrödinger equations with variable coefficients are investigated by utilizing the bifurcation theory and the complete discrimination system method. Firstly, coupled nonlinear Schrödinger equations with variable coefficients are transformed into coupled nonlinear ordinary differential equations by the traveling wave transformations. Then, phase portraits of coupled nonlinear Schrödinger equations with variable coefficients are plotted by selecting the suitable parameters. Furthermore, the traveling wave solutions of coupled nonlinear Schrödinger equations with variable coefficients which correspond to phase orbits are easily obtained by applying the method of planar dynamical systems, which can help us to further understand the propagation of the coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optics. Finally, the periodic wave solutions, implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions of the coupled nonlinear Schrödinger equations with variable coefficients are constructed.

Solution of coupled nonlinear Schrödinger equations in focusing-defocusing medium by modified perturbation theory

The interaction of bright solitons of different orders and two different wavelengths propagating in the medium focusing for one wavelength and defocusing for the other is considered. The system of nonlinear Schrödinger equations is solved by means of perturbation theory. Application of an additional postulate to adjust both widths of the solitons and to modify the amplitude by a factor determined by the overlap integral greatly improves the accuracy of the description. The good accuracy of description is confirmed by numerical calculations. Full Text: PDF ReferencesY. Kivshar, G. P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals, (Amsterdam, Academic Press 2003). CrossRef F. Abdullaev, S. Darmanyan, P. Khabibullaev, Optical Solitons, (Springer-Verlag, Berlin, 1993) CrossRef G.I.A Stegema, D.N. Christodoulides, M. Segev, IEEE J. Selected Topics Quantum Electron. 6, (2000), 1419 CrossRef J. Yang, "Nonlinear Waves in Integrable and Nonintegrable Systems", (SIAM, Philadelphia 2010). CrossRef Y. Kivshar, B. Malomed, "Dynamics of solitons in nearly integrable systems", Rev. Mod. Phys. 61, 763 (1989). CrossRef P.G. Kevrekidis, D.J. Frantzeskakis, "Solitons in coupled nonlinear Schrödinger models: A survey of recent developments", Reviews in Physics 1 (2016), 140 CrossRef R. de la Fuente, A. Barthelemy, "Spatial soliton-induced guiding by cross-phase modulation", IEEE J. Quantum Electron. 28, 547 (1992). CrossRef H. T. Tran, R. A. Sammut, "Families of multiwavelength spatial solitons in nonlinear Kerr media", Phys. Rev. A 52, 3170 (1995). CrossRef S. Leble, B. Reichel, "Coupled nonlinear Schrödinger equations in optic fibers theory", Eur. Phys. J. Special Topics 173, 5 (2009). CrossRef M. Vijayajayanthi, T.Kanna, M. Lakshmanan, "Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities", Eur. Phys. J. Special Topics 173, 57 (2009). CrossRef S. V. Manakov, "On the theory of two-dimensional stationary self-focusing of electromagnetic waves ", Sov. Phys. JETP 38 (1973), 248 DirectLink J. Yang, Phys. Rev. E 65, 036606 (2002). CrossRef T.Kanna, M. Lakshmanan, "Exact Soliton Solutions, Shape Changing Collisions, and Partially Coherent Solitons in Coupled Nonlinear Schrödinger Equations", Phys. Rev. Lett. 86, 5043 (2001). CrossRef M. Jakubowski, K. Steiglitz, R. Squier, "State transformations of colliding optical solitons and possible application to computation in bulk media", Phys. Rev. E 58, 6752 (1998). CrossRef P. S. Jung, W. Krolikowski, U. A. Laudyn, M. Trippenbach, and M. A. Karpierz, "Supermode spatial optical solitons in liquid crystals with competing nonlinearities", Phys. Rev. A 95 (2017). CrossRef P. S. Jung, M. A. Karpierz, M. Trippenbach, D. N. Christodoulides, and W. Krolikowski, "Supermode spatial solitons via competing nonlocal nonlinearities", Photonics Lett. Pol. 10 (2018). CrossRef A. Ramaniuk, M. Trippenbach, P.S. Jung, D.N. Christodoulides, W.Krolikowski, G. Assanto, "Scalar and vector supermode solitons owing to competing nonlocal nonlinearities", Opt. Express 29, 8015 (2021) CrossRef

Collision-induced amplitude dynamics of fast 2D solitons in the presence of the generic nonlinear loss

We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schrödinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of perturbed solitons and the second step, which is the crucial one, is for the analysis of the collision-induced change in the envelope of the perturbed 2D soliton. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding-solitons. In addition, we show that the current perturbative technique can be simply applied to study the collision-induced amplitude shift in a fast collision of two perturbed 1D solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations in the presence of the cubic loss and in the presence of the quintic loss.

ANALYTICAL SOLUTIONS OF COUPLED NONLINEAR SCHRODINGER EQUATIONS FOR TWO NON LINEARLY INTERACTING STOKES WAVE TRAINS OVER INFINITE DEPTH

An attempt to find the exact analytical solutions of the two coupled nonlinear Schrodinger equations of 3rd order occurring from the oblique interaction of two capillary gravity wave trains in the case of crossing sea states in deep water is the main premise of the present paper. The solutions obtained here are due to the nonlinear interaction of two Stokes wave trains in one spatial dimension. Graphs have been plotted to investigate the influence of capillarity on the amplitudes of such wave trains. From 3D figures it has been observed that the capillarity has diminishing influence on the amplitudes of the either wave packet.