Symmetry classification and optimal systems of a non-linear wave equation

In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.     

On the nonlocal symmetries of certain nonlinear oscillators and their general solution

In this Letter we establish the integrability of two nonlinear oscillators through group theoretical method. We utilize the algorithm given in [M.L. Gandarias, M.S. Bruzon, J. Nonlinear Math. Phys. 18 (2011) 123] and construct nonlocal symmetries for these two oscillators. From the knowledge of the latter we derive first integral and general solution for these two nonlinear nonpolynomial oscillator equations.     

Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations

The Type-II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. In [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622] Abraham-Shrauner and Govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries. The Lie point symmetries of a model equation and the two-dimensional Burgers' equation and their descendants were used to identify the hidden symmetries. In this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries. We have considered the same models presented in [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622], as well as the WDVV equations of associativity in two-dimensional topological field theory which reduces, in the case of three fields, to a single third order equation of Monge-Ampère type. We have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using Lie symmetries, however when is reduced by using nonclassical symmetries the reduced partial differential equation gains Lie symmetries.     

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.     

Weak self-adjointness and conservation laws for a porous medium equation

The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006, 2007) [4], [7]. In Ibragimov (2007) [6] a general theorem on conservation laws was proved. In Gandarias (2011) [3] we generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In this paper we find the subclasses of weak self-adjoint porous medium equations. By using the property of weak self-adjointness we construct some conservation laws associated with symmetries of the differential equation.     

Similarity reductions of a nonlinear model for vibrations of beams

In this paper we find exact solutions for a beam equation. We make a full analysis of the symmetry reductions of this equation by using the classical Lie method of infinitesimals. We present some explicit solutions: compacton solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.     

Hydrodeoxygenation of the Angelica Lactone Dimer,a Cellulose‐Based Feedstock: Simple,High‐Yield Synthesis of Branched C7–C10 Gasoline‐like Hydrocarbons

Dehydration of biomass‐derived levulinic acid under solid acid catalysis and treatment of the resulting angelica lactone with catalytic K₂CO₃ produces the angelica lactone dimer in excellent yield. This dimer serves as a novel feedstock for hydrodeoxygenation, which proceeds under relatively mild conditions with a combination of oxophilic metal and noble metal catalysts to yield branched C₇–C₁₀ hydrocarbons in the gasoline volatility range. Considering that levulinic acid is available in >80 % conversion from raw biomass, a field‐to‐tank yield of drop‐in, cellulosic gasoline of >60 % is possible.