Division theorem combined with the Riccati equation for solving some nonlinear Schrödinger-like equations

2012 ◽  
Vol 219 (4) ◽  
pp. 1686-1694
Author(s):  
Bin Lu
Keyword(s):  
Riccati Equation ◽  
Nonlinear Schrödinger ◽  
Division Theorem

scholarly journals Solitons: Conservation laws and dressing methods

2019 ◽  
Vol 34 (06n07) ◽  
pp. 1930003
Author(s):  
Anastasia Doikou ◽  
Iain Findlay
Keyword(s):  
Conservation Laws ◽  
Riccati Equation ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Soliton Solutions ◽  
Toda Chain ◽  
Lax Pair ◽  
Conserved Quantities ◽  
Nonlinear Schrödinger ◽  
Marchenko Equation

We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux–Bäcklund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov–Shabat dressing scheme and the Gelfand–Levitan–Marchenko equation are also introduced. Via this method, generic solutions are produced and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete nonlinear Schrödinger model, the Korteweg–de Vries and nonlinear Schrödinger equations as well as the sine-Gordon and Liouville models.

Solitons and other solutions to the coupled nonlinear Schrödinger type equations

2017 ◽  
Vol 6 (2) ◽  
Author(s):  
M. M. El-Borai ◽  
H. M. El-Owaidy ◽  
Hamdy M. Ahmed ◽  
A. H. Arnous ◽  
M. Mirzazadeh
Keyword(s):  
Nonlinear Optics ◽  
Plasma Physics ◽  
Riccati Equation ◽  
Deep Water ◽  
Water Waves ◽  
Solution Method ◽  
Periodic Waves ◽  
Trial Solution ◽  
Nonlinear Schrödinger ◽  
Integration Schemes

AbstractNonlinear Schrödinger type equations arise from a wide variety of fields, such as fluids, nonlinear optics, the theory of deep water waves, plasma physics, and so on. In this paper, two integration schemes are employed to obtain solitons, periodic waves and other forms of solutions of the coupled nonlinear Schrödinger type equations. The two schemes that are studied in this paper are the Bäcklund transformation of Riccati equation and the trial solution method.

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