The nonlinear superposition theorem of Lie and Abel’s differential equations

1983 ◽  
Vol 38 (13) ◽  
pp. 448-452 ◽  
Author(s):  
J. L. Reid ◽  
G. L. Strobel
Keyword(s):  
Differential Equations ◽  
Nonlinear Superposition ◽  
Superposition Theorem

scholarly journals THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

1998 ◽  
Vol 13 (21) ◽  
pp. 3601-3627 ◽  
Author(s):  
J. F. CARIÑENA ◽  
G. MARMO ◽  
J. NASARRE
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Order Differential Equation ◽  
Superposition Principle ◽  
Theoretical Perspective ◽  
Nonlinear Superposition ◽  
Theoretical Methods ◽  
First Order ◽  
Second Order Differential Equations

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

scholarly journals REDUCED DYNAMICS FROM THE UNITARY GROUP TO SOME FLAG MANIFOLDS: INTERACTING MATRIX RICCATI EQUATIONS

2009 ◽  
Vol 06 (04) ◽  
pp. 573-581 ◽  
Author(s):  
KAZUYUKI FUJII ◽  
HIROSHI OIKE
Keyword(s):  
Differential Equations ◽  
Riccati Equation ◽  
Time Evolution ◽  
Unitary Group ◽  
Natural Generalization ◽  
Flag Manifolds ◽  
Level System ◽  
Nonlinear Superposition ◽  
N Level ◽  
Nonlinear Superposition Formula

In this paper we treat the time evolution of unitary elements in the N level system and consider the reduced dynamics from the unitary group U(N) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies. Our result is a natural generalization of the paper Chaturvedi et al. [1] (arXiv: 0706.0964 [quant-ph]).

scholarly journals On Differential Equations Derived from the Pseudospherical Surfaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hongwei Yang ◽  
Xiangrong Wang ◽  
Baoshu Yin
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Curvature Scalar ◽  
Riemann Curvature ◽  
Tensor Fields ◽  
Nonlinear Superposition ◽  
Metric Tensor ◽  
New Exact Solutions ◽  
Pseudospherical Surfaces

We construct two metric tensor fields; by means of these metric tensor fields, sinh-Gordon equation and elliptic sinh-Gordon equation are obtained, which describe pseudospherical surfaces of constant negative Riemann curvature scalarσ= −2,σ= −1, respectively. By employing the Bäcklund transformation, nonlinear superposition formulas of sinh-Gordon equation and elliptic sinh-Gordon equation are derived; various new exact solutions of the equations are obtained.

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