An Efficient Method for Analytically Solving (1+2)-dimensional Chiral Non-linear Schrödinger Equation

In this paper, we have successfully extracted novel analytic solutions for the (1+2)-dimensional Chiral non-linear Schrödinger (NLS) equation by modified extended tanh expansion method combined with new Riccati solutions (METEM-cNRCS) as far as we know. When a wave transformation is applied to the considered Chiral NLS equation, a nonlinear ODE is obtained. Assuming the solutions of ODE have a form as the method suggests, and substituting the trial solutions to the ODE, we get a polynomial. Gathering the coefficients with the same power in the polynomial, we acquire an algebraic equation system. So, we may obtain the abundant solutions of the (1+2)-dimensional Chiral NLS equation by solving the system via Maple. The plots of some solutions are demonstrated to explain the dynamics of the solutions. It is expected that the results of the paper are a guide for future works in traveling wave theory.

Modulation Instability of Hydro-Elastic Waves Blown by a Wind with a Uniform Vertical Profile

An interesting physical phenomenon was recently observed when a fresh-water basin is covered by a thin ice film that has properties similar to the property of a rubber membrane. Surface waves can be generated under the action of wind on the air–water interface that contains an ice film. The modulation property of hydro-elastic waves (HEWs) in deep water covered by thin ice film blown by the wind with a uniform vertical profile is studied here in terms of the airflow velocity versus wavenumber. The modulation instability of HEWs is studied through the analysis of coefficients of the nonlinear Schrödinger (NLS) equation with the help of the Lighthill criterion. The NLS equation is derived using the multiple scale method in the presence of airflow. It is demonstrated that the potentially unstable hydro-elastic waves with negative energy appear for relatively small wind speeds, whereas the Kelvin–Helmholtz instability arises when the wind speed becomes fairly strong. Estimates of parameters of modulated waves for the typical conditions are given.

Modulation Instability of Hydro-Elastic Waves Blown by a Wind with a Uniform Vertical Profile

An interesting physical phenomenon was recently observed when a fresh-water basin is covered by a thin ice film that has properties similar to that of a rubber membrane. Surface waves can be generated under the action of wind on the air-water interface that contains an ice film. The modulation property of hydro-elastic waves (HEWs) in deep water covered by thin ice film blown by the wind with a uniform vertical profile is studied here in terms of the air-flow velocity versus a wavenumber. The modulation instability of HEWs is studied through the analysis of coefficients of the nonlinear Schrödinger (NLS) equation with the help of the Lighthill criterion. The NLS equation is derived using the multiple scale method in the presence of airflow. It is demonstrated that the potentially unstable hydro-elastic waves with negative energy appear for relatively small wind speeds, whereas the Kelvin–Helmholtz instability arises when the wind speed becomes fairly strong. Estimates of parameters of modulated waves for the typical conditions are given.

Analytical simulations of the Fokas system; extension (2 + 1)-dimensional nonlinear Schrödinger equation

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

Elliptic- and hyperbolic-function solutions of the nonlocal reverse-time and reverse-space-time nonlinear Schrödinger equations

In this paper, we obtain the stationary elliptic- and hyperbolic-function solutions of the nonlocal reverse-time and reverse-space-time nonlinear Schrödinger (NLS) equations based on their connection with the standard Weierstrass elliptic equation. The reverse-time NLS equation possesses the bounded dn-, cn-, sn-, sech-, and tanh-function solutions. Of special interest, the tanh-function solution can display both the dark- and antidark-soliton profiles. The reverse-space-time NLS equation admits the general Jacobian elliptic-function solutions (which are exponentially growing at one infinity or display the periodical oscillation in x), the bounded dn- and cn-function solutions, as well as the K-shifted dn- and sn-function solutions. At the degeneration, the hyperbolic-function solutions may exhibit an exponential growth behavior at one infinity, or show the gray- and bright-soliton profiles.

Soliton Waves in Lossy Nonlinear Transmission Lines at Microwave Frequencies: Analytical, Numerical and Experimental Studies

In this paper, we performed analytical, numerical and experimental studies on the generation of soliton waves in discrete nonlinear transmission lines (NLTL) with varactors, as well as the analysis of the losses impact on the propagation of these waves. Using the reductive perturbation method, we derived a nonlinear Schrödinger (NLS) equation with a loss term and determined an analytical expression that completely describes the bright soliton profile. Our theoretical analysis predicts the carrier wave frequency threshold above which a formation of bright solitons can be observed. We also performed numerical simulations to confirm our analytical results and we analyzed the space–time evolution of the soliton waves. A good agreement between analytical and numerical findings was obtained. An experimental prototype of the lossy NLTL, built at the discrete level, was used to validate our proposed model. The experimental shape of the envelope solitons is well fitted by the theoretical waveforms, which take into account the amplitude damping due to the losses in commercially available varactors and inductors used in a prototype. Experimentally observed changes in soliton amplitude and half–maximum width during the propagation along lossy NLTL are in good accordance with the proposed model defined by NLS equation with loss term.