A nonlinear superposition principle admitted by coupled Riccati equations of the projective type

1980 ◽  
Vol 4 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Robert L. Anderson
Keyword(s):  
Superposition Principle ◽  
Riccati Equations ◽  
Nonlinear Superposition ◽  
Coupled Riccati Equations ◽  
Projective Type

scholarly journals Waveless subcritical flow past symmetric bottom topography

2012 ◽  
Vol 24 (2) ◽  
pp. 213-230 ◽  
Author(s):  
R. J. HOLMES ◽  
G. C. HOCKING ◽  
L. K. FORBES ◽  
N. Y. BAILLARD
Keyword(s):  
Nonlinear Problem ◽  
Parameter Space ◽  
Bottom Topography ◽  
Superposition Principle ◽  
Trapped Waves ◽  
Special Shape ◽  
Nonlinear Superposition ◽  
Subcritical Flow ◽  
Flow Past

The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

scholarly journals Bäcklund Transformation of Fractional Riccati Equation and Infinite Sequence Solutions of Nonlinear Fractional PDEs

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bin Lu
Keyword(s):  
Partial Differential Equations ◽  
Riccati Equation ◽  
Infinite Sequence ◽  
Bäcklund Transformation ◽  
Superposition Principle ◽  
Fractional Partial Differential Equations ◽  
Backlund Transformation ◽  
Nonlinear Superposition ◽  
Fractional Pdes ◽  
Liouville Derivative

The Bäcklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided.

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