## Direct Reduction and Differential Constraints (1994)

Citations: | 28 - 2 self |

### BibTeX

@MISC{Olver94directreduction,

author = {Peter Olver},

title = {Direct Reduction and Differential Constraints},

year = {1994}

}

### OpenURL

### Abstract

. Direct reductions of partial differential equations to systems of ordinary differential equations are in one-to-one correspondence with compatible differential constraints. The differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions, but admits a noncharacteristic second order reduction if and only if it is linearizable. 1. Introduction. One of the most useful methods for determining particular explicit solutions to partial differential equations is to reduce them to ordinary differential equations (which are presumably easier to solve) through a suitably inspired ansatz. The classical Lie method for finding group-invariant solutions, first described in full generality in [7], generalizes and includes well-known methods for finding similarity solutions, travelling wave solutions, and other basic reduction methods. For example, the solutions which are invariant under a one-parameter symmetry...

### Citations

1056 |
Application of Lie Groups to Differential Equations
- Olver
- 1986
(Show Context)
Citation Context ...wide variety of equations, the required calculations being very reminiscent of those needed to solve the determining equations for the symmetry groups of the system of partial differential equations, =-=[13]-=-. 2. Compatibility and Reduction. For the most part, we will concentrate on the simple case of a single second order partial differential equations in two independent and one dependent variables, but ... |

119 |
Digerential Algebra
- Ritt
- 1950
(Show Context)
Citation Context ...imilarity variables (group invariants) reduces the equation to an ordinary differential equation. Methods based on the Riquier–Ritt theory of overdetermined systems of partial differential equations, =-=[16]-=-, and differential Gröbner bases, [8], can then, at least in principle, be effectively used to analyze these systems, and thereby determine classes of useful, compatible differential constraints. This... |

81 |
The symmetry approach to classification of integrable equations
- Mikhăılov, Shabat, et al.
- 1991
(Show Context)
Citation Context ...h possesses a generalized symmetry of order five or more, or even a formal symmetry of rank five or more (and, in particular, a recursion operator), is necessarily equivalent to such an equation; see =-=[10]-=-. I do not know whether there is a deep connection between the higher order direct reduction method and the existence of formal symmetries and/or recursion operators. For example, an interesting specu... |

75 |
The general similarity solution of the heat equation
- Bluman, Cole
- 1969
(Show Context)
Citation Context ...solutions which are invariant under a one-parameter symmetry group of a partial differential equation in two independent variables can be found by solving a reduced ordinary differential equation. In =-=[1]-=-, Bluman and Cole extended Lie’s reduction method to include nonclassical symmetry groups, where the invariance of the partial differential equation is only required on its intersection with the invar... |

43 |
Differential Gröbner Bases
- Mansfield
- 1991
(Show Context)
Citation Context ... reduces the equation to an ordinary differential equation. Methods based on the Riquier–Ritt theory of overdetermined systems of partial differential equations, [16], and differential Gröbner bases, =-=[8]-=-, can then, at least in principle, be effectively used to analyze these systems, and thereby determine classes of useful, compatible differential constraints. This paper is devoted to a systematic stu... |

35 |
Non-classical symmetry reduction: example of the Boussinesq equation
- Levi, Winternitz
- 1989
(Show Context)
Citation Context ...ed and subsequently applied to many partial differential equations arising in a wide variety of physical systems. It was then realized that the direct approach is included in the nonclassical method, =-=[6]-=-, although the latter slightly more general owing to a restriction on the type of ansatz allowed, [12]. More recently, in a study of blow-up of solutions to nonlinear diffusion equations, Galaktionov,... |

33 | The construction of special solutions to partial differential equations, Phys. Lett. A 114
- Olver, Rosenau
- 1986
(Show Context)
Citation Context ...n functions. One can readily envision extending Galaktionov’s method to even more unknown functions, although at present I am unaware of any significant examples. Earlier, Philip Rosenau and I, [14], =-=[15]-=-, showed how (almost) all known reduction methods, including the classical and nonclassical methods, partial invariance, separation of variables, etc., can be placed into a general framework. The gene... |

31 |
New similarity reductions of the Boussinesq equation
- Clarkson, Kruskal
- 1989
(Show Context)
Citation Context ... condition characterizing the group-invariant functions. An alternative † Supported in part by NSF Grant DMS 91–16672 and DMS 92–04192. 1sdirect reduction method was proposed by Clarkson and Kruskal, =-=[2]-=-, where a systematic approach for determining ansätze which reduce the partial differential equation to a single ordinary differential equation was developed and subsequently applied to many partial d... |

28 |
Group invariant solutions of differential equations
- Olver, Ph
- 1987
(Show Context)
Citation Context ...unknown functions. One can readily envision extending Galaktionov’s method to even more unknown functions, although at present I am unaware of any significant examples. Earlier, Philip Rosenau and I, =-=[14]-=-, [15], showed how (almost) all known reduction methods, including the classical and nonclassical methods, partial invariance, separation of variables, etc., can be placed into a general framework. Th... |

27 | Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities - Galaktionov - 1995 |

24 |
On new exact blow-up solutions for nonlinear heat conduction equations with source and applications, Differential Integral Equations 3
- Galaktionov
- 1990
(Show Context)
Citation Context ... although the latter slightly more general owing to a restriction on the type of ansatz allowed, [12]. More recently, in a study of blow-up of solutions to nonlinear diffusion equations, Galaktionov, =-=[3]-=-, proposed a generalization of the direct method, which he called the method of “nonlinear separation”, in which the ansatz involves two different functions of the similarity variable, and the partial... |

12 |
Linearization of second order ordinary differential equations via Cartan’s equivalence method
- Grissom, Wilkens
- 1989
(Show Context)
Citation Context ...int to the simple form u yy = 0 through a clever change of coordinates. (This is because not every second order ordinary differential equation can be linearized by a point transformation; see [17] and=-=[5]-=- for explicit necessary and sufficient conditions for effecting such a linearization.) Theorem 4. The ansatz (16) will reduce the partial differential equation (1) to a coupled system of n distinct or... |

10 |
The Nonclassical Method is More General than the Direct Method for Symmetry Reductions. An Example of the FitzhughNagumo Equation, Phys
- NUCCI, CLARKSON
- 1992
(Show Context)
Citation Context ...cal systems. It was then realized that the direct approach is included in the nonclassical method, [6], although the latter slightly more general owing to a restriction on the type of ansatz allowed, =-=[12]-=-. More recently, in a study of blow-up of solutions to nonlinear diffusion equations, Galaktionov, [3], proposed a generalization of the direct method, which he called the method of “nonlinear separat... |

8 |
Z ur allgemeinen Theorie der partiellen Differentialgleichungen beliebeger
- Lie
(Show Context)
Citation Context ...differential equations (which are presumably easier to solve) through a suitably inspired ansatz. The classical Lie method for finding group-invariant solutions, first described in full generality in =-=[7]-=-, generalizes and includes well-known methods for finding similarity solutions, travelling wave solutions, and other basic reduction methods. For example, the solutions which are invariant under a one... |

6 |
Differential constraints and one-parameter groups of Lie-Bäcklund transformations
- Meleshko
- 1983
(Show Context)
Citation Context ...sulting overdetermined system of partial differential equations satisfy some form of compatibility condition. Special cases of this approach appear in the earlier work of Yanenko, [20], and Meleshko, =-=[9]-=-, and more recent extensions appear in the work of Vorob’ev, [18], [19]. For example, in the classical and nonclassical methods, the differential constraint is the invariant surface condition, and its... |

6 |
Reduction and quotient equations for differential equations with symmetries
- Vorob’ev
- 1991
(Show Context)
Citation Context ...satisfy some form of compatibility condition. Special cases of this approach appear in the earlier work of Yanenko, [20], and Meleshko, [9], and more recent extensions appear in the work of Vorob’ev, =-=[18]-=-, [19]. For example, in the classical and nonclassical methods, the differential constraint is the invariant surface condition, and its compatibility with the partial differential equation implies tha... |

5 |
Détermination des Invariants ponctuels de l’ Équation differentielle ordinaire de second ordre: y ′′ = w(x, y, y ′). Preisschriften der fürstlichen Jablonowski’schen Gesellschaft XXXII
- Tresse
- 1896
(Show Context)
Citation Context ... constraint to the simple form u yy = 0 through a clever change of coordinates. (This is because not every second order ordinary differential equation can be linearized by a point transformation; see =-=[17]-=- and[5] for explicit necessary and sufficient conditions for effecting such a linearization.) Theorem 4. The ansatz (16) will reduce the partial differential equation (1) to a coupled system of n dist... |

3 |
Symmetries of compatibility conditions for systems of differential equations
- Vorob'ev
- 1992
(Show Context)
Citation Context ...y some form of compatibility condition. Special cases of this approach appear in the earlier work of Yanenko, [20], and Meleshko, [9], and more recent extensions appear in the work of Vorob’ev, [18], =-=[19]-=-. For example, in the classical and nonclassical methods, the differential constraint is the invariant surface condition, and its compatibility with the partial differential equation implies that the ... |

2 | Functional separation of variables for Laplace equations in two dimensions
- Miller, Rubel
- 1993
(Show Context)
Citation Context ...linear separation ansatz, namely u = U(w 1 (z 1 )+w 2 (z 2 )) where w 1 and w 2 are functions of different similarity variables z 1 = ζ 1 (x, t), z 2 = ζ 2 (x, t), was considered by Miller and Rubel, =-=[11]-=-, in a recent analysis of Laplace–Beltrami equations on Riemannian surfaces. These and similar reductions are governed by particular types of third order differential constraints. Finally, we investig... |

2 |
Theory of consistency and methods of integrating systems of nonlinear partial differential equations
- Yanenko
- 1964
(Show Context)
Citation Context ...k), such that the resulting overdetermined system of partial differential equations satisfy some form of compatibility condition. Special cases of this approach appear in the earlier work of Yanenko, =-=[20]-=-, and Meleshko, [9], and more recent extensions appear in the work of Vorob’ev, [18], [19]. For example, in the classical and nonclassical methods, the differential constraint is the invariant surface... |