C. Muriel

Integrating Factors and λ −Symmetries

Abstract We investigate the relationship between integrating factors and λ–symmetries for ordinary differential equations of arbitrary order. Some results on the existence of λ–symmetries are used to prove an independent existence theorem for integrating factors. A new method to calculate integrating factors and the associated first integrals is derived from the method to compute λ–symmetries and the associated reduction algorithm. Several examples illustrate how the method works in practice and how the computations that appear in other methods may be simplified.

Nonlocal transformations and linearization of second-order ordinary differential equations

The class of nonlinear second-order equations that are linearizable by means of generalized Sundman transformations (S-transformations) is identified as the class of equations admitting first integrals that are polynomials of first degree in the first-order derivative. This class is also characterized in terms of the coefficients of the equations and constructive methods to derive the linearizing S-transformations are presented. Only the equations of a well-defined subclass can also be linearized by invertible point transformations. These invertible point transformations can be constructed by using the algorithms for the calculation of linearizing S-transformations. Several examples illustrate that both types of linearization are strictly different.

SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS AND FIRST INTEGRALS OF THE FORM A(t, x)ẋ + B(t, x)

We characterize the equations in the class of the second-order ordinary differential equations ẍ = M(t, x, ẋ) which have first integrals of the form A(t, x)ẋ + B(t, x). We give an intrinsic characterization of the equations in and an algorithm to calculate explicitly such first integrals. Although includes equations that lack Lie point symmetries, the equations in do admit λ-symmetries of a certain form and can be characterized by the existence of such λ-symmetries. The equations in a well-defined subclass of can completely be integrated by using two independent first integrals of the form A(t, x)ẋ + B(t, x). The methods are applied to several relevant families of equations.

New Symmetry Reductions for some Ordinary Differential Equations

Abstract In this work we derive potential symmetries for ordinary differential equations. By using these potential symmetries we find that the order of the ODE can be reduced even if this equation does not admit point symmetries. Moreover, in the case for which the ODE admits a group of point symmetries, we find that the potential symmetries allow us to perform further reductions than its point symmetries. Some diffusion equations admitting an infinite number of potential symmetries and the scaling group as a Lie symmetry are considered and some general results are obtained. For all the equations that we have studied, a set of potential symmetries admitted by the diffusion equation is “inherited” by the ODE that emerges as the reduced equation under the scaling group.

Prolongations of vector fields and the invariants-by-derivation property

For any given vector field X defined on some open set M ⊂∝2, we characterize the prolongations Xn* of X to the nth jet space M(n), n≥1, such that a complete system of invariants for Xn* can be obtained by derivation of lower-order invariants. This leads to characterizations of C∞-symmetries and to new procedures for reducing the order of an ordinary differential equation.

THE CALOGERO-BOGOYAVLENSKII-SCHIFF EQUATION IN 2+1 DIMENSIONS

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.

The Schwarzian Korteweg–de Vries equation in (2 + 1) dimensions

In this paper, the complete Lie group classification of a (2 + 1)-dimensional integrable Schwarzian Korteweg–de Vries equation is obtained. The reduction to systems of partial differential equations in (1 + 1) dimension is derived from the optimal system of subalgebras. The invariance study of these systems leads to second-order ODEs. These ODEs provide several classes of solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the second and third Painleve transcendents. The corresponding solutions of the (2 + 1)-dimensional equation involve up to three arbitrary smooth functions. They even appear in the form ρ(z)f(x + (t)). Consequently, the solutions exhibit a rich variety of qualitative behaviour. Indeed, by making appropriate choices for the arbitrary functions, we are able to exhibit large families of solitary waves, coherent structures and different types of bound states.

Integrability of equations admitting the nonsolvable symmetry algebra so(3, R)

If an ordinary differential equation admits the nonsolvable Lie algebra so(3,R), and we use any of its generators to reduce the order, the reduced equation does not inherit the remaining symmetries. We prove here how the lost symmetries can be recovered as C∞-symmetries of the reduced equation. If the order of the last reduced equation is higher than one, these C∞-symmetries can be used to obtain new order reductions. As a consequence, a classification of the third-order equations that admit so(3,R) as symmetry algebra is given and a step-by-step method to solve the equations is presented.

SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH FIRST INTEGRALS OF THE FORM C(t) + 1/(A(t, x)ẋ + B(t, x))

We study the class of the ordinary differential equations of the form ẍ + a2(t, x)ẋ2 + a1(t, x)ẋ + a0(t, x) = 0, that admit v = ∂x as λ-symmetry for some λ = α(t, x)ẋ + β(t, x). This class coincides with the class of the second-order equations that have first integrals of the form C(t) + 1/(A(t, x)ẋ + B(t, x)). We provide a method to calculate the functions A, B and C that define the first integral. Some relationships with the class of equations linearizable by local and a specific type of nonlocal transformations are also presented.

Traveling-wave solutions of the Schwarz-Korteweg-de Vries equation in 2+1 dimensions and the Ablowitz-Kaup-Newell-Segur equation through symmetry reductions

One of the more interesting solutions of the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in 2+1 dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in 2+1 dimensions. It is interesting that neither of the (2+1)-dimensional integrable systems considered admit Virasoro-type subalgebras.