We consider solutions of the non-linear von Neumann equation involving Jacobi’s elliptic functions sn, cn, and dn, and 3 linearly independent operators. In two cases one can construct a state-dependent Hamiltonian which is such that the corresponding non-linear von Neumann equation is solved by the given density operator. We prove that in a certain context these two cases are the only possibili...

The usual non-linear corrections for a Helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations. We aim to present a systematic derivation of a solution of the non-linear Helmholtz resonator equation, in order to obtain analytically expressions for impedances close to resonance, while including non-linear effects. The amplitude r...

The local and non-local vector Non-linear Schrodinger Equation (NLSE) with a general cubic non-linearity are considered in presence of linear term characterized, general, by non-hermitian matrix which under certain condition incorporates balanced loss gain coupling between the complex fields governing non-linear equations. It is shown that systems posses Lax pair an infinite number conserved qu...

Solution to unsteady Navier-Stokes equation by Splitting method in physical orthogonal algebraic curvilinear coordinate system, also termed ‘Non-linear grid system’ is presented. The linear terms in Navier-Stokes equation are solved by CrankNicholson method while the non-linear term is solved by the second order Adams-Bashforth method. This work is meant to bring together the advantage of Split...

solving a large sparse set of linear equations is of the problems widely seen in every numerical investigation in the entire range of engineering disciplines. employing a finite element approach in solving partial derivative equations, the resulting stiffness matrices would contain many zero-valued elements. moreover, storing all these sparse matrices in a computer memory would slower the compu...

For k ≥ 3 let A ⊂ [1, N ] be a set not containing a solution to a1x1 + · · · + akxk = a1xk+1 + · · · + akx2k in distinct integers. We prove that there is an ε > 0 depending on the coefficients of the equation such that every such A has O(N) elements. This answers a question of I. Ruzsa. Introduction The problems of estimating the size of a largest subset of [1, N ] not containing a solution to ...

We propose an extension of Walsh’s classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a dri...

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as t→±∞ (x/t∼O(1)) of the solution to the Cauchy problem for the modified non-linear Schrödinger equation, i∂tu+ 1 2 ∂ x u+ |u|u+is∂x(|u|u) = 0, s ∈ R>0, which is a model for non-linear pulse propagation in optical fibres...

Electro-active solids are solids that are either infused with electrorheological fluids or embedded with electrically conducting particles, the body as a whole however conducting negligible current. In this paper, we provide a mathematical framework, within the context of continuum mechanics, for the study of electro-active solids. The theory assumes that the body can be considered as a continu...