scholarly journals INTEGRABILITY OF THE RICCATI EQUATION FROM A GROUP-THEORETICAL VIEWPOINT

1999 ◽  
Vol 14 (12) ◽  
pp. 1935-1951 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
ARTURO RAMOS
Keyword(s):  
Riccati Equation ◽  
Superposition Principle ◽  
Theoretical Perspective ◽  
Nonlinear Superposition ◽  
Theoretical Methods ◽  
Integrability Conditions

In this paper we develop some group-theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation, and we discuss some of its integrability conditions from a group-theoretical perspective. The nonlinear superposition principle also arises in a simple way.

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 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.

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 Measure valued solutions of asymptotically homogeneous semilinear hyperbolic systems in one space variable

1990 ◽  
Vol 33 (3) ◽  
pp. 443-460 ◽  
Author(s):  
F. Demengel ◽  
J. Rauch
Keyword(s):  
Initial Data ◽  
Weak Convergence ◽  
Radon Measure ◽  
Space Variable ◽  
Hyperbolic Systems ◽  
Superposition Principle ◽  
Nonlinear Superposition ◽  
Image Position ◽  
Semilinear Hyperbolic Systems ◽  
Characteristic Coordinates

We study systems which in characteristic coordinates have the formwhere A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.Simple weak converge of the initial data does not imply weak convergence of the solutions.

On the periodic solution of the Burgers equation: a unified approach

1992 ◽  
Vol 438 (1902) ◽  
pp. 113-132 ◽  
Keyword(s):  
Periodic Solution ◽  
Ad Hoc ◽  
Burgers Equation ◽  
Superposition Principle ◽  
Theta Functions ◽  
Periodic Problem ◽  
Periodic Wave ◽  
Sine Wave ◽  
Nonlinear Superposition ◽  
Riemann Theta Functions

For the most part, the various known expressions for the periodic solution to the Burgers equation have been derived by a mixture of ad hoc argument and approximation methods. These include the well-known solutions due to R. D. Fay and D. F. Parker. Remarkably, these solutions were obtained without recourse to the celebrated Hopf-Cole transformation which linearizes the Burgers equation (to the heat conduction equation). We present here an alternative approach to the periodic problem which uses the fact that the Riemann theta functions ‘solve’ the Burgers equation via the Hopf-Cole transformation, and thereby provide a unifying framework in which all previous representations of the periodic solution find their natural place. Yet another expression for the periodic solution, in the form of an amplitude modulated sine wave, is given and would appear to be new. The limiting forms of the periodic solution as t → 0 (embryonic ‘sawtooth’ profile) and t →∞ (‘old-age’ sinusoidal profile) are examined in detail. The representation due to D. F. Parker, which expresses the periodic wave as a superposition of spreading Taylor shock profiles, is discussed within the context of a nonlinear superposition principle . The composite asymptotic expansion due to M. B. Lesser and D. G. Crighton is examined in the light of our results. The curious behaviour of this expansion in the embryonic region ( t ≪ 1), whereby a first-order approximation yields an exact periodic solution (the Fay solution), is explained.

scholarly journals SEMI-CLASSICAL WAVE PACKET DYNAMICS FOR HARTREE EQUATIONS

2011 ◽  
Vol 23 (09) ◽  
pp. 933-967 ◽  
Author(s):  
PEI CAO ◽  
RÉMI CARLES
Keyword(s):  
Initial Data ◽  
Classical Limit ◽  
Wave Packets ◽  
Superposition Principle ◽  
Approximate Solutions ◽  
Wave Functions ◽  
Nonlinear Superposition ◽  
Wave Packet Dynamics ◽  
Classical Wave ◽  
Ehrenfest Time

We study the propagation of wave packets for nonlinear nonlocal Schrödinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.

Sign in / Sign up

Export Citation Format

Share Document