Using certain finite-dimensional stable range of the nonlinear terms, we obtain large families of exact solutions parameterized by functions for the equation of nonstationary transonic gas flows discovered by Lin, Reisner and Tsien, and its three-dimensional generalization.

I discuss a recent application of homotopy perturbation and Adomian decomposition methods to the linear and nonlinear Schrödinger equations. I propose a generalization of the procedure for the treatment of a wider class of problems.

We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau’s theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions...

In this paper, we investigate the rupture dynamics of thin liquid films driven by intermolecular forces via weakly nonlinear bifurcation analysis. The dynamic equations governing slow dynamics of the perturbation amplitude of the near-critical mode corresponding to several models describing the evolution of thin liquid films in different physical situations appear to have the same structure. Wh...

In this paper, a methodology to obtain approximate models from input-output data for nonlinear, causal, time invariant discrete systems having a certain type of continuity condition called fading memory is presented. The region or domain of the input space, where the model can be applicable, is studied, as well ss the importance of this study in applications as data processing and the qualifica...

Existing dictionary learning algorithms are based on the assumption that the data are vectors in an Euclidean vector space ℝ d , and the dictionary is learned from the training data using the vector space structure of ℝ d and its Euclidean L2-metric. However, in many applications, features and data often originated from a Riemannian manifold that does not support a global linear (vector space) ...

We consider an integrable generalization of the nonlinear Schrödinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Deri...

Let Kn ̄2x `2n :x i & 0 for 1% i% n ́ and suppose that f :Kn MNKn is nonexpansive with respect to the F " -norm and f(0) ̄ 0. It is known that for every x `Kn there exists a periodic point ξ ̄ ξ x `Kn (so f p(ξ ) ̄ ξ for some minimal positive integer p ̄ pξ) and f k(x) approaches 2 f j(ξ ) :0% j! p ́ as k approaches infinity. What can be said about P*(n), the set of positive integers p for which there...

Concerning the problem of shape and pattern description, a compact formula (Gielis’ formula) has recently been proposed that generates a vast diversity of natural shapes. However, this formula is a modified version of the equation for the circle, so that it is expressed in terms of trigonometric functions which are inherent to linear phenomena but rarely appear in the description of nonlinear p...

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial diierential equations. Using the introduced deenition of reduction of diierential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equat...