Analytical methods via bright–dark solitons and solitary wave solutions of the higher-order nonlinear Schrödinger equation with fourth-order dispersion

2019 ◽  
Vol 33 (35) ◽  
pp. 1950443 ◽  
Author(s):  
Aly R. Seadawy ◽  
Mujahid Iqbal ◽  
Dianchen Lu
Keyword(s):  
Solitary Wave ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Solitary Wave Solutions ◽  
Physical Sciences ◽  
Dark Solitons ◽  
Wave Solutions ◽  
Order Dispersion

In this research work, we investigated the higher-order nonlinear Schrödinger equation (NLSE) with fourth-order dispersion, self-steepening, nonlinearity, nonlinear dispersive terms and cubic-quintic terms which is described as the propagation of ultra-short pulses in fiber optics. We apply the modification form of extended auxiliary equation mapping method to find the new exact and solitary wave solutions of higher-order NLSE. As a result, new solutions are obtained in the form of solitons, kink–anti-kink type solitons, bright–dark solitons, traveling wave, trigonometric functions, elliptic functions and periodic solitary wave solutions. These new different types of solutions show the power and fruitfulness of this new method and also show two- and three-dimensional graphically with the help of computer software Mathematica. These new solutions have many applications in the field of physics and other branches of physical sciences. We can also solve other higher-order nonlinear partial differential equations (NPDEs) involved in mathematical physics and other various branches of physical sciences with this new technique.

OPTICAL SOLITARY WAVE SOLUTIONS FOR THE FOURTH-ORDER DISPERSIVE CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION

2007 ◽  
Vol 21 (15) ◽  
pp. 2657-2668 ◽  
Author(s):  
CHAO-QING DAI ◽  
JUN-LANG CHEN ◽  
JIE-FANG ZHANG
Keyword(s):  
Solitary Wave ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Solitary Wave Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Quintic Nonlinear Schrödinger Equation

We present several kinds of optical solitary wave solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation describing the propagation of optical pulses in a medium that exhibits a parabolic nonlinearity law. Among them, apart from some regular fundamental bright solitary wave solutions and dark solitary wave solutions given, there are also two kinds of combined solitary wave solutions, i.e., bright and dark solitary wave and W-shaped solitary wave solutions which describe bright and dark solitary wave properties. It is found that their amplitude may approach nonzero when the time variable approaches infinity. Moreover, when the parameters of the bright and dark solitary wave are selected appropriately, the kink and anti-kink solitary wave solutions, which are first reported in the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation, are discovered.

Analytical Dark Solitary Wave Solutions for the Higher Order Nonlinear Schrödinger Equation with Cubic-quintic Terms

2000 ◽  
Vol 55 (3-4) ◽  
pp. 397-400 ◽  
Author(s):  
Woo-pyo Hong
Keyword(s):  
Solitary Wave ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Phase Method ◽  
Solitary Wave Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

By means of the coupled amplitude-phase method we find analytical dark solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short (femtosecond) optical soliton propagation in non-Kerr media. The dark solitary wave solution exists even for the coefficients of quintic terms much larger than those of cubic terms.

scholarly journals New Solitary-wave Solutions for the Higher Order Nonlinear Schrödinger Equation with Both Real and Imaginary Raman Terms

2003 ◽  
Vol 58 (12) ◽  
pp. 667-671 ◽  
Author(s):  
Woo-Pyo Hong
Keyword(s):  
Solitary Wave ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Optical Fibers ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Solitary Wave Solutions ◽  
Nonlinear Schrödinger ◽  
Wave Solutions

We find new solitary-wave solutions of the higher order nonlinear Schrödinger equation with both real and imaginary Raman terms, which can model an ultrashort pulse propagation through optical fibers, under some constraint among the model coefficients. The physical conditions such as the wavelength needed to launch the pulse, the types of optical fibers, and the required peak power, are obtained from the constraints for the solitary-wave solutions. - PACS number(s): 42.65.Tg, 42.81Dp, 02.30.Jr, 42.79.Sz.

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