A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebr...

where the index α gets summed over α “ 0, 1, . . . , 3, and the indices J,K are summed over 1, . . . ,N. Equations of this type play a prominent role in the theory of nonlinear waves, due to the special cancellation structure inherent in them, namely the Q0-null-form. This causes equations of this type to be amenable to better global regularity results than generic quadratic source terms, as wi...

The invariance of nonlinear partial differential equations under a certain infinite-dimensional Lie algebra AN (z) in N spatial dimensions is studied. The special case A1(2) was introduced in J. Stat. Phys. 75, 1023 (1994) and contains the Schrödinger Lie algebra sch1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of AN...

Using a modified WKB approach, we present a rigorous semi-classical analysis for solutions of nonlinear Schrödinger equations with rotational forcing. This yields a rigorous justification for the hydrodynamical system of rotating superfluids. In particular it is shown that global-in-time semi-classical convergence holds whenever the limiting hydrodynamical system has global smooth solutions and...

We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalising the one introduced by Squier for word rewriting systems. We characterise this property by using the notion of critical branching. In particular, we define sufficient conditions f...

Most Lie group integrators can be expanded in series indexed by the set of ordered rooted trees. To each tree one can associate two distinct higher order derivation operators, which we call frozen and unfrozen operators. Composition of frozen operators induces a concatenation product on the trees, whereas composition of unfrozen operators induces a somewhat more complicated product known as the...

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group G are an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions ...

Lie group analysis is a powerful tool for obtaining exact similarity solutions of nonlinear (integro-) differential equations. In order to calculate the group-invariant solutions one first has to find the full Lie point symmetry group admitted by the given (integro-)differential equations and to determine all the subgroups of this Lie group. An effective, systematic means to classify the simila...