scholarly journals Exact Solutions to the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Xin-Lei Mai ◽  
Wei Li ◽  
Shi-Hai Dong
Keyword(s):  
Exact Solutions ◽  
Optical Fibers ◽  
Time Dependent ◽  
Jacobi Elliptic Function ◽  
Optical Pulses ◽  
Nonlinear Schrödinger ◽  
Ultrashort Optical Pulses ◽  
Trial Function Method ◽  
High Order Time ◽  
Special Case

In this paper, a trial function method is employed to find exact solutions to the nonlinear Schrödinger equations with high-order time-dependent coefficients. This system might be used to describe the propagation of ultrashort optical pulses in nonlinear optical fibers, with self-steepening and self-frequency shift effects. The new general solutions are found for the general case a 0 ≠ 0 including the Jacobi elliptic function solutions, solitary wave solutions, and rational function solutions which are presented in comparison with the previous ones obtained by Triki and Wazwaz, who only studied the special case a 0 = 0 .

scholarly journals Nonlinear Schrödinger-type formulation of scalar field cosmology: Two barotropic fluids and exact solutions

2020 ◽  
Vol 35 (19) ◽  
pp. 2050157
Author(s):  
Chonticha Kritpetch ◽  
Jarunee Sanongkhun ◽  
Pichet Vanichchapongjaroen ◽  
Burin Gumjudpai
Keyword(s):  
Scalar Field ◽  
Exact Solutions ◽  
Time Dependent ◽  
Frw Cosmology ◽  
Dependent Case ◽  
Nonlinear Schrödinger ◽  
Barotropic Fluids ◽  
Scalar Field Cosmology ◽  
Formulation Variables ◽  
Canonical Scalar Field

Time-independent nonlinear Schrödinger-type (NLS) formulation of FRW cosmology with canonical scalar field is considered in the case of two barotropic fluids. We derived Friedmann formulation variables in terms of NLS variables. Seven exact solutions found by D’Ambroise [Ph.D. thesis, arXiv:1005.1410 ] and one new found solution are explored and tested in cosmology. The result suggests that time-independent NLS formulation of cosmology case should be upgraded to the time-dependent case.

scholarly journals A Generalized Kundu-Eckhaus Equation With An Extra-Dispersion: Pulses Configuration

2021 ◽  
Author(s):  
Hamdy. Abdel-Gawad
Keyword(s):  
Exact Solutions ◽  
Optical Fibers ◽  
Phase Modulation ◽  
Raman Effect ◽  
Optical Pulses ◽  
Spectral Content ◽  
Sharp Waves ◽  
The Unified Method ◽  
Unified Method ◽  
Cascade Complex

Abstract Raman effect is due to self-phase modulation (SPM), which is embedded in Kundu-Eckhaus equation KEE. Here, a generalized KEE is suggested by accounting for an extra dispersion. Here, we are concerned with finding the exact solutions of the proposed equation, which is done by using the unified method. In this work, we aim to show that the optical pulses OPs propagation in optical fibers may show a variety of shapes. Waves of multiple geometric shapes are observed. Among these waves, hybrid lumps, soliton, cascade, complex chirped, hybrid w-shaped, rhombus (diamond) waves and soliton modulation, which is induced by SPM. Further, the pulses intensity, frequency, wavelength, polarization, and spectral content are introduced. The results found here are of great interest in experimenting the effects of the induced dispersion on pulses configurations. Further, the colliding dynamics are inspected and as it is observed that no rogue or sharp waves formation holds, so the collision is elastic.

Novel solitons and elliptic function solutions of (1 + 1)-dimensional higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms and its applications

2019 ◽  
Vol 33 (22) ◽  
pp. 1950253
Author(s):  
Muhammad Arshad ◽  
Aly R. Seadawy ◽  
Dianchen Lu ◽  
Abdullah
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Elliptic Function ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Mapping Technique ◽  
Jacobi Elliptic Function ◽  
Nonlinear Schrödinger

The propagations are generally described through nonlinear Schrödinger equation (NLSE) in the optical solitons. In the NLSEs, the higher order NLSE with derivative non-Kerr nonlinear terms is a model that depicts propagation of pulses beyond ultra-short range in optical communication system. Several novel exact solutions of different kinds such as solitons, solitary waves and Jacobi elliptic function solutions are achieved via using modified extended mapping technique. Different kinds of exact results have prestigious exertions in engineering and physics. Structures of solitons different kinds are shown graphically by giving suitable values to parameters. The physical interpretations of solutions can be understand through structures. Several exact solutions and computing work confirm the supremacy and usefulness of the current technique.

scholarly journals Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Li-hua Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Time Dependent ◽  
Group Method ◽  
New Exact Solutions ◽  
Petviashvili Equation ◽  
The Kdv Equation ◽  
Special Case

The (2 + 1)-dimensional Kadomtsev-Petviashvili equation with time-dependent coefficients is investigated. By means of the Lie group method, we first obtain several geometric symmetries for the equation in terms of coefficient functions and arbitrary functions oft. Based on the obtained symmetries, many nontrivial and time-dependent conservation laws for the equation are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, the (2 + 1)-dimensional KP equation is reduced to (1 + 1)-dimensional nonlinear partial differential equations, including a special case of (2 + 1)-dimensional Boussinesq equation and different types of the KdV equation. At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions.

Exact Solutions for N-Coupled Nonlinear Schrödinger Equations With Variable Coefficients

2016 ◽  
Vol 71 (7) ◽  
pp. 665-672 ◽  
Author(s):  
Bo Tang ◽  
Yingzhe Fan ◽  
Jixiu Wang ◽  
Shijun Chen
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Nonlinear Schrödinger Equations ◽  
Jacobi Elliptic Function ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractIn this paper, based on similarity transformation and auxiliary equation method, we construct many exact solutions ofN-coupled nonlinear Schrödinger equations with variable coefficients, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions. These solutions may give insight into many considerable physical processes.

scholarly journals Bright, dark, and singular optical soliton solutions for perturbed Gerdjikov-Ivanov equation

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 151-156
Author(s):  
Esma Ulutas ◽  
Mustafa Inc ◽  
Dumitru Baleanu ◽  
Sunil Kumar
Keyword(s):  
Exact Solutions ◽  
Optical Fibers ◽  
Elliptic Function ◽  
Soliton Solutions ◽  
Elliptic Functions ◽  
Optical Soliton ◽  
Jacobi Elliptic Function ◽  
Jacobi Elliptic Functions ◽  
Ansatz Method ◽  
Pulse Dynamics

This study consider Gerdjikov-Ivanov equation where the perturbation terms appear with full non-linearity. The Jacobi elliptic function ansatz method is implemented to obtain exact solutions of this equation that models pulse dynamics in optical fibers. It is retrieved some bright, dark optical and singular solitons profile in the limiting cases of the Jacobi elliptic functions. The constraint conditions depending on the parameters for the existence of solitons are also presented.

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