scholarly journals Spatiotemporal Complexity Mediated by Higher-Order Peregrine-Like Extreme Events

2021 ◽  
Vol 9 ◽  
Author(s):  
Saliya Coulibaly ◽  
Camus G. L. Tiofack ◽  
Marcel G. Clerc
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Analytical Description ◽  
Rogue Waves ◽  
Dissipative Systems ◽  
Largest Lyapunov Exponent ◽  
Dissipative Effects ◽  
Non Linear ◽  
The Stability ◽  
Good Agreement

The Peregrine soliton is the famous coherent solution of the non-linear Schrödinger equation, which presents many of the characteristics of rogue waves. Usually studied in conservative systems, when dissipative effects of injection and loss of energy are included, these intrigued waves can disappear. If they are preserved, their role in the dynamics is unknown. Here, we consider this solution in the framework of dissipative systems. Using the paradigmatic model of the driven and damped non-linear Schrödinger equation, the profile of a stationary Peregrine-type solution has been found. Hence, the Peregrine soliton waves are persistent in systems outside of the equilibrium. In the weak dissipative limit, analytical description has a good agreement with the numerical simulations. The stability has been studied numerically. The large bursts that emerge from the instability are analyzed by means of the local largest Lyapunov exponent. The observed spatiotemporal complexity is ruled by the unstable second-order Peregrine-type soliton.

scholarly journals Extended (G'/G) Method Applied to the Modified Non-Linear Schrodinger Equation in the Case of Ocean Rogue Waves

2014 ◽  
Vol 04 (04) ◽  
pp. 246-256
Author(s):  
Atock A. Nwatchok Stéphane ◽  
Daika Augustin ◽  
Mbane Biouélé César
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Waves ◽  
Non Linear

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

Hybrid high algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives

2016 ◽  
Vol 27 (05) ◽  
pp. 1650049 ◽  
Author(s):  
Junyan Ma ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Truncation Error ◽  
Resonance Problem ◽  
Phase Lag ◽  
Step Method ◽  
Algebraic Order ◽  
Test Equation ◽  
The Stability ◽  
Scalar Test Equation

A hybrid tenth algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives are obtained in this paper. We will investigate • the construction of the method • the local truncation error (LTE) of the newly obtained method. We will also compare the lte of the newly developed method with other methods in the literature (this is called the comparative LTE analysis) • the stability (interval of periodicity) of the produced method using frequency for the scalar test equation different from the frequency used in the scalar test equation for phase-lag analysis (this is called stability analysis) • the application of the newly obtained method to the resonance problem of the Schrödinger equation. We will compare its effectiveness with the efficiency of other known methods in the literature. It will be proved that the developed method is effective for the approximate solution of the Schrödinger equation and related periodical or oscillatory initial value or boundary value problems.

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