scholarly journals Stability of small periodic waves for the nonlinear Schrödinger equation

2007 ◽  
Vol 234 (2) ◽  
pp. 544-581 ◽  
Author(s):  
Thierry Gallay ◽  
Mariana Hărăguş
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Periodic Waves ◽  
Nonlinear Schrödinger ◽  
Small Periodic

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.

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 Instability of Double-Periodic Waves in the Nonlinear Schrödinger Equation

2021 ◽  
Vol 9 ◽  
Author(s):  
Dmitry E. Pelinovsky
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Periodic Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Linear Equations ◽  
Nonlinear Schrodinger Equation ◽  
Periodic Waves ◽  
Nls Equation ◽  
Nonlinear Schrödinger

It is shown how to compute the instability rates for the double-periodic solutions to the cubic NLS (nonlinear Schrödinger) equation by using the Lax linear equations. The wave function modulus of the double-periodic solutions is periodic both in space and time coordinates; such solutions generalize the standing waves which have the time-independent and space-periodic wave function modulus. Similar to other waves in the NLS equation, the double-periodic solutions are spectrally unstable and this instability is related to the bands of the Lax spectrum outside the imaginary axis. A simple numerical method is used to compute the unstable spectrum and to compare the instability rates of the double-periodic solutions with those of the standing periodic waves.

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