scholarly journals Rogue periodic waves of the focusing nonlinear Schrödinger equation

2018 ◽  
Vol 474 (2210) ◽  
pp. 20170814 ◽  
Author(s):  
Jinbing Chen ◽  
Dmitry E. Pelinovsky
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Elliptic Functions ◽  
Nonlinear Schrodinger Equation ◽  
Periodic Waves ◽  
Jacobian Elliptic Functions ◽  
Nonlinear Schrödinger

Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrödinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn . Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov–Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine’s breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.

NEW ENVELOPE SOLUTIONS FOR COMPLEX NONLINEAR SCHRÖDINGER+ EQUATION VIA SYMBOLIC COMPUTATION

2003 ◽  
Vol 14 (02) ◽  
pp. 225-235 ◽  
Author(s):  
ZHEN-YA YAN
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Symbolic Computation ◽  
Schrodinger Equation ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Nonlinear Schrodinger Equation ◽  
Jacobian Elliptic Functions ◽  
Nonlinear Schrödinger ◽  
Trigonometric Function Solutions

With the aid of symbolic computation, an extended Jacobian elliptic function expansion method is further extended to the complex nonlinear Schrödinger + equation. As a result, 24 families of the envelope doubly-periodic solutions with Jacobian elliptic functions are obtained. When the modulus m → 1 or zero, the corresponding six envelope solitary wave solutions and six envelope singly-periodic (trigonometric function) solutions are also found. This powerful method can also be applied to other equations, such as the nonlinear Schrödinger equation and Zakharov equation.

scholarly journals Chirped Periodic and Solitary Waves for Improved Perturbed Nonlinear Schrödinger Equation with Cubic Quadratic Nonlinearity

2021 ◽  
Vol 5 (4) ◽  
pp. 234
Author(s):  
Aly R. Seadawy ◽  
Syed T. R. Rizvi ◽  
Saad Althobaiti
Keyword(s):  
Schrödinger Equation ◽  
Fiber Optics ◽  
Nonlinear Schrödinger Equation ◽  
Solitary Waves ◽  
Schrodinger Equation ◽  
Elliptic Functions ◽  
Nonlinear Schrodinger Equation ◽  
Quadratic Nonlinearity ◽  
Periodic Waves ◽  
Nonlinear Schrödinger

In this paper, we study the improved perturbed nonlinear Schrödinger equation with cubic quadratic nonlinearity (IPNLSE-CQN) to describe the propagation properties of nonlinear periodic waves (PW) in fiber optics. We obtain the chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF) and also obtain some solitary waves (SW) such as dark, bright, hyperbolic, singular and periodic solitons. The nonlinear chirp associated with each of these optical solitons was observed to be dependent on the pulse intensity. The graphical behavior of these waves will also be displayed.

scholarly journals Exact Solutions of the Generalized Nonlinear Schrodinger Equation

2021 ◽  
pp. 18-25
Author(s):  
Gaukhar Shaikhova ◽  
Arailym Syzdykova ◽  
Samgar Daulet
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Physical Problems ◽  
Different Types ◽  
Generalized Nonlinear Schrödinger Equation

In this work, the generalized nonlinear Schrodinger equation is investigated. Exact solutions are derived by the sinecosine method. This method is used to obtain the exact solutions for different types of nonlinear partial differential equations. Graphs of obtained solutions are presented. The obtained solutions are found to be important for the explanation of some practical physical problems.

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