Solutions of solitary-wave for the variable-coefficient nonlinear Schrödinger equation with two power-law nonlinear terms

2018 ◽  
Vol 32 (28) ◽  
pp. 1850310 ◽  
Author(s):  
Le Xin ◽  
Ying Kong ◽  
Lijia Han
Keyword(s):  
Solitary Wave ◽  
Power Law ◽  
Optical Lattice ◽  
Variable Coefficient ◽  
Transformation Method ◽  
Potential Space ◽  
Nls Equation ◽  
Quadratic Potential ◽  
Nonlinear Schrödinger ◽  
Constant Coefficients

In this paper, we consider the variable-coefficient power-law nonlinear Schrödinger equations (NLSEs) with external potential as well as the gain or loss function. First, we generalize the similarity transformation method, which converts the variable coefficient NLSE with two power-law nonlinear terms to the autonomous dual-power NLS equation with constant coefficients. Then, we obtain the exact solutions of the variable-coefficient NLSE. Moreover, we discuss the solitary-wave solutions for equations with vanishing potential, space-quadratic potential and optical lattice potential, which are applied to many branches of physics.

A Direct Transformation Method and its Application to Variable Coefficient Nonlinear Equations of Schrödinger Type

2009 ◽  
Vol 64 (11) ◽  
pp. 697-708 ◽  
Author(s):  
Li-Hua Zhang ◽  
Xi-Qiang Liu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Transformation Method ◽  
Nonlinear Evolution Equations ◽  
Nls Equation ◽  
Direct Transformation ◽  
Nonlinear Schrödinger

In this paper, the generalized variable coefficient nonlinear Schrödinger (NLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients are directly reduced to simple and solvable ordinary differential equations by means of a direct transformation method. Taking advantage of the known solutions of the obtained ordinary differential equations, families of exact nontravelling wave solutions for the two equations have been constructed. The characteristic feature of the direct transformation method is, that without much extra effort, we circumvent the integration by directly reducing the variable coefficient nonlinear evolution equations to the known ordinary differential equations. Another advantage of the method is that it is independent of the integrability of the given nonlinear equation. The method used here can be applied to reduce other variable coefficient nonlinear evolution equations to ordinary differential equations.

A variable coefficient nonlinear Schrödinger equation: localized waves on the plane wave background and their dynamics

2019 ◽  
Vol 33 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Variable Coefficient ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Plane Wave Background ◽  
Localized Waves ◽  
Similarity Reductions

In this work, a variable coefficient nonlinear Schrödinger (vc-NLS) equation is under investigation, which can describe the amplification or absorption of pulses propagating in an optical fiber with distributed dispersion and nonlinearity. By means of similarity reductions, a similar transformation helps us to relate certain class of solutions of the standard NLS equation to the solutions of integrable vc-NLS equation. Furthermore, we analytically consider nonautonomous breather wave, rogue wave solutions and their interactions in the vc-NLS equation, which possess complicated wave propagation in time and differ from the usual breather waves and rogue waves. Finally, the main characteristics of the rational solutions are graphically discussed. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves.

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

2016 ◽  
Vol 30 (10) ◽  
pp. 1650106 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jian Chen
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Variable Coefficients ◽  
Higher Order ◽  
Optical Systems ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Order Variable ◽  
Generalized Darboux Transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

Nonautonomous Solitons for the Coupled Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equations with External Potentials in the Non-Kerr Fibre

2015 ◽  
Vol 70 (12) ◽  
pp. 985-994
Author(s):  
Bing-Qing Mao ◽  
Yi-Tian Gao ◽  
Yu-Jie Feng ◽  
Xin Yu
Keyword(s):  
Variable Coefficient ◽  
Velocity Dispersion ◽  
Group Velocity Dispersion ◽  
Soliton Solutions ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Nonlinear Schrödinger ◽  
Similarity Transformations ◽  
Constant Coefficients ◽  
Bose Einstein

AbstractVariable-coefficient nonlinear Schrödinger (NLS)-type models are used to describe certain phenomena in plasma physics, nonlinear optics, arterial mechanics, and Bose–Einstein condensation. In this article, the coupled variable-coefficient cubic-quintic NLS equations with external potentials in the non-Kerr fibre are investigated. Via symbolic computation, similarity transformations and relevant constraints on the coefficient functions are obtained. Based on those transformations, such equations are transformed into the coupled cubic-quintic NLS equations with constant coefficients. Nonautonomous soliton solutions are derived, and propagation and interaction of the nonautonomous solitons in the non-Kerr fibre are investigated analytically and graphically. Those soliton solutions are related to the group velocity dispersion r(x) and external potentials h1(x) and h2(x, t). With the different choices of r(x), parabolic, cubic, and periodically oscillating solitons are obtained; with the different choices of h2(x, t), we can see the dromion-like structures and nonautonomous solitons with smooth step-like oscillator frequency profiles, to name a few.

scholarly journals Dynamics of Different Nonlinearities to the Perturbed Nonlinear Schrödinger Equation via Solitary Wave Solutions with Numerical Simulation

2021 ◽  
Vol 5 (4) ◽  
pp. 213
Author(s):  
Asim Zafar ◽  
Muhammad Raheel ◽  
Muhammad Qasim Zafar ◽  
Kottakkaran Sooppy Nisar ◽  
Mohamed S. Osman ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Solitary Wave ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Power Law ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Solitary Wave Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

This paper investigates the solitary wave solutions for the perturbed nonlinear Schrödinger equation with six different nonlinearities with the essence of the generalized classical derivative, which is known as the beta derivative. The aforementioned nonlinearities are known as the Kerr law, power, dual power law, triple power law, quadratic–cubic law and anti-cubic law. The dark, bright, singular and combinations of these solutions are retrieved using an efficient, simple integration scheme. These solutions suggest that this method is more simple, straightforward and reliable compared to existing methods in the literature. The novelty of this paper is that the perturbed nonlinear Schrödinger equation is investigated in different nonlinear media using a novel derivative operator. Furthermore, the numerical simulation for certain solutions is also presented.

Solutions of nonlinear Schrödinger equation for interfacial waves propagating between two ideal fluids

2009 ◽  
Vol 87 (6) ◽  
pp. 675-684 ◽  
Author(s):  
A. M. Abourabia ◽  
M. A. Mahmoud ◽  
G. M. Khedr
Keyword(s):  
Dispersion Relation ◽  
Multiple Scale ◽  
Transformation Method ◽  
Wave Number ◽  
Sine Gordon Equation ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Inviscid Fluids ◽  
Ideal Fluids ◽  
Infinite Extent

We present solutions to the problem of waves propagating at an interface between two inviscid fluids of infinite extent and differing densities. The method of multiple scale is employed to obtain a dispersion relation and nonlinear Schrödinger (NLS) equation, which describes the behavior of the system for the fluid interface. The dispersion relation of the model NLS equation is studied. The solutions of the NLS equation are derived analytically by using the complex tanh-function method and the function transformation method into a sine-Gordon equation. Also, diagrams are drawn to illustrate the elevation of the interface, the slip velocity, and the conservation of power. We observe that the elevation of the interface is in the form of traveling quasi-solitary waves that decrease as the wave number increases. We see that the slip velocities also bring a nonlinear and periodic characters. Finally, we observe that the conservation of power is in the form of traveling waves. Also, as the wave number increases, the conservation of power is more accurate in fluctuating around zero.

scholarly journals ON INTEGRABILITY OF VARIABLE COEFFICIENT NONLINEAR SCHRÖDINGER EQUATIONS

2012 ◽  
Vol 24 (07) ◽  
pp. 1250015 ◽  
Author(s):  
CİHANGİR ÖZEMİR ◽  
FARUK GÜNGÖR
Keyword(s):  
Exact Solutions ◽  
Theoretical Approach ◽  
Variable Coefficient ◽  
Analytic Solutions ◽  
Painlevé Test ◽  
Nls Equation ◽  
Lax Pairs ◽  
Soliton Theory ◽  
Nonlinear Schrödinger ◽  
Point Transformations

We apply Painlevé test to the most general variable coefficient nonlinear Schrödinger (VCNLS) equations as an attempt to identify integrable classes and compare our results versus those obtained by the use of other tools like group-theoretical approach and the Lax pairs technique of the soliton theory. We presented some exact solutions based on point transformations relating analytic solutions of VCNLS equations for specific choices of the coefficients to those of the standard integrable NLS equation.

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