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The purpose of this paper is to construct analytical solutions of the (2+1)-dimensional nonlinear Schrodinger equations, and the existence of rogue waves and their localized structures are studied.

Function transformation and variable separation method are applied to the (2+1)-dimensional nonlinear Schrodinger equations.

A series of analytical solutions including rogue wave solutions for the (2+1)-dimensional nonlinear Schrodinger equations are constructed. Localized structures of rogue waves confirm the presence of large amplitude wave wall.

The localized structures of rogue waves are displayed by analytical solutions and figures. The authors just find some of them and others still to be found.

These results may help to investigate the localized structures and the behavior of rogue waves for nonlinear evolution equations. Applying two different function transformations and variable separation functions to two different states of the equations, respectively, to construct the solutions of the (2+1)-dimensional nonlinear Schrodinger equations. Rogue wave solutions are enumerated and their figures are partly displayed.

The nonlinear Schrödinger equation plays a vital role in wave mechanics and nonlinear optics. The purpose of this paper is the fractal paradigm of the nonlinear Schrödinger equation for the calculation of novel solitary solutions through the variational principle.

Appropriate traveling wave transform is used to convert a partial differential equation into a dimensionless nonlinear ordinary differential equation that is handled by a semi-inverse variational technique.

This paper sets out the Schrödinger equation fractal model and its variational principle. The results of the solitary solutions have shown that the proposed approach is very accurate and effective and is almost suitable for use in such problems.

Nonlinear Schrödinger equation is an important application of a variety of various situations in nonlinear science and physics, such as photonics, the theory of superfluidity, quantum gravity, quantum mechanics, plasma physics, neutron diffraction, nonlinear optics, fiber-optic communication, capillary fluids, Bose–Einstein condensation, magma transport and open quantum systems.

The variational principle of the Schrödinger equation without the Lagrange multiplier method in the sense of the fractal calculus is developed for the first time in the literature to the best of the author's understanding.

The purpose of this paper is to study the stability analysis and optical solitary wave solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation, which are derived from a multicomponent plasma with nonextensive distribution.

Based on the ansatz and sub-equation theories, the authors use a direct method to find stability analysis and optical solitary wave solutions of the (2 + 1)-dimensional equation.

By considering the ansatz method, the authors successfully construct the bright and dark soliton solutions of the equation. The sub-equation method is also extended to find its complexitons solutions. Moreover, the explicit power series solution is also derived with its convergence analysis. Finally, the influences of each parameter on these solutions are discussed via graphical analysis.

The dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional nonlinear Schrödinger equation type nonlinear wave fields.

The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which can be used to describe the propagation properties of optical soliton solutions.

The authors apply the ansatz method and the Hamiltonian system technique to find its bright, dark and Gaussian wave solitons and analyze its modulation instability analysis and stability analysis solution.

The results imply that the generalized nonlinear Schrödinger equation has bright, dark and Gaussian wave solitons. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Some constraint conditions are provided which can guarantee the existence of solitons. The authors analyze its modulation instability analysis and stability analysis solution.

These results may help us to further study the local structure and the interaction of solutions in generalized nonlinear Schrödinger -type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of the generalized nonlinear Schrödinger--type equations.

The purpose of this paper is to present a reliable treatment of the Fokas–Lenells equation, an integrable generalization of the nonlinear Schrödinger equation. The authors use a special complex envelope traveling-wave solution to carry out the analysis. The study confirms the accuracy and efficiency of the used method.

The proposed technique, namely, the trial equation method, as presented in this work has been shown to be very efficient for solving nonlinear equations with spatio-temporal dispersion.

A class of chirped soliton-like solutions including bright, dark and kink solitons is derived. The associated chirp, including linear and nonlinear contributions, is also determined for each of these optical pulses. Parametric conditions for the existence of chirped soliton solutions are presented.

The paper presents a new efficient algorithm for handling an integrable generalization of the nonlinear Schrödinger equation.

The authors present a useful algorithm to handle nonlinear equations with spatial-temporal dispersion. The method is an effective method with promising results.

This is a newly examined model. A useful method is presented to offer a reliable treatment.

The paper presents a new efficient algorithm for handling an integrable generalization of the nonlinear Schrödinger equation.

The purpose of this paper is to consider the time-splitting Fourier spectral (TSFS) method to solve the fractional coupled Klein–Gordon–Schrödinger (K-G-S) equations. A time-splitting spectral approach is applied for discretizing the Schrödinger-like equation and along with that, a pseudospectral discretization has been accurately utilized for the temporal derivatives in the Klein–Gordon-like equation. Furthermore, the time-splitting scheme is proved to be unconditionally stable. Numerical experiments guarantee high accuracy of the TSFS scheme for the K-G-S equations. Here, the derivative of fractional order is taken in the Riesz sense.

The focus of this paper is to study the Riesz fractional coupled K-G-S equations using the TSFS method. This method is dependent on evaluating the solution to the given problem in small steps, and treating the nonlinear and linear steps separately. The nonlinear step is made in the time domain, while the linear step is made in the frequency domain, which necessitates the use of Fourier transform back and forth. It is a very effective, powerful and efficient method to solve the nonlinear differential equations, as in previous works (Bao et al., 2002; Bao and Yang, 2007; Muslu and Erbay, 2003; Borluk et al., 2007), the initial and boundary-value problem is decomposed into linear and nonlinear subproblems. Summarizing the technique of the TSFS method, it can be stated that first the Schrödinger-like equation is solved in two splitting steps. Then, the Klein–Gordon-like equation is solved by discretizing the spatial derivatives by means of the pseudospectral method.

The utilized method is found to be very efficient and accurate. Moreover, the time-splitting spectral scheme is found to be unconditionally stable. By means of thorough study, it is found that the spectral method is time-reversible, is gauge-invariant and also conserves the total charge. Moreover, the results have been graphically presented to exhibit the accuracy of the proposed methods. Apart from that, the numerical solutions have been also compared with the exact solutions. Numerical experiments establish that the proposed technique manifests high accuracy and efficiency.

To the authors’ best knowledge, the Riesz fractional coupled K-G-S equations have been for the first time solved by using the TSFS method.

The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme.

The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities.

The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients.

The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.

This paper seeks to develop, propose and validate, through a series of presentable examples, a comprehensive high‐precision and ultra‐fast computing concept for solving stiff ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN).

The core of the concept developed in this paper is a straight‐forward scheme that we call “nonlinear adaptive optimization (NAOP)”, which is used for a precise template calculation for solving any (stiff) nonlinear ODEs through CNN processors.

One of the key contributions of this work (this is a real breakthrough) is to demonstrate the possibility of mapping/transforming different types of nonlinearities displayed by various classical and well‐known oscillators (e.g. van der Pol‐, Rayleigh‐, Duffing‐, Rössler‐, Lorenz‐, and Jerk‐ oscillators, just to name a few) unto first‐order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN‐templates. Furthermore, in case of PDEs solving, the same concept also allows a mapping unto first‐order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDEs in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultra‐fast solver of stiff differential equations (both ODEs and PDEs). This clearly enables a CNN‐based, real‐time, ultra‐precise, and low‐cost Computational Engineering. As proof of concept a well‐known prototype of stiff equations (van der Pol) has been considered; the corresponding precise CNN‐templates are derived to obtain precise solutions of this equation.

This paper contributes to the enrichment of the literature as the relevant state‐of‐the‐art does not provide a systematic and robust method to solve nonlinear ODEs and/or nonlinear PDEs using the CNN‐paradigm. Further, the “NAOP” concept developed in this paper has been proven to perform accurate and robust calculations. This concept is not based on trial‐and‐error processes as it is the case for various classes of optimization methods/tools (e.g. genetic algorithm, particle swarm, neural networks, etc.). The “NAOP” concept developed in this frame does significantly contribute to the consolidation of CNN as a universal and ultra‐fast solver of nonlinear differential equations (both ODEs and PDEs). An implantation of the concept developed is possible even on embedded digital platforms (e.g. field‐programmable gate array (FPGA), digital signal processing (DSP), graphics processing unit (GPU), etc.); this opens a broad range of applications. On‐going works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting PDE models such as Navier Stokes, Schrödinger, Maxwell, etc.

The purpose of this paper is to employ an alternative numerical approach to analyze the characteristics of superconducting charge qubit based on a single Cooper pair box (SCB), also to study the influence of the bias current.

The paper starts with the circuit model of a charge qubit system based on Josephson junction using Hamiltonian formalism. Corresponding Schrodinger eigenvalue problem with periodic boundary condition is converted to the Mathieu type eigenvalue problem. By applying finite difference technique, energy spectrum of charge qubit is obtained and the solutions in the lowest band are obtained in the form of Bloch waves whose superposition provides a wave packet to investigate the effect of bias current to the Coulomb blockade.

The paper identifies a periodic tridiagonal Hermitian matrix form of the eigenvalue problem that is believed to be a special eigenvalue problem. The paper emphasizes that Schrodinger formalism is very useful to model superconducting qubits systems. The investigations indicate that the bias current strongly influences the Coulomb blockade and expectation value of supercurrent (as well as number of Cooper pairs) are affected by gate voltage and energy scale.

In the present study, Schrodinger eigenvalue problem is time independent, therefore, current‐voltage characteristics of the charge qubit system could not be considered. The solution technique applied here can also be used to apply other type of Josephson junction based qubits and circuits.

The paper includes theoretical findings for the development of superconducting qubit that can be valuable for experimentalist. The result obtained in this study is useful for the comparison of experimental study with the expectation value of number of Cooper pairs as function of gate voltage. Working parameters of a SCB can be determined from the findings.

This paper fulfils the contribution of the numerical study of Schrodinger equation for the investigation of superconducting qubits under the influence of bias current.