We study the matrix equation X+A>X−1A = Q, where A is a complex square matrix and Q is complex symmetric. Special cases of this equation appear in Green’s function calculation in nano research and also in the vibration analysis of fast trains. In those applications, the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators. The s...

The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report...

in this paper, we obtained the 1-soliton solutions of the symmetric regularized long wave (srlw) equation and the (3+1)-dimensional shallow water wave equations. solitary wave ansatz method is used to carry out the integration of the equations and obtain topological soliton solutions the physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. note t...

The vacuum solution ds = dx + x dy + 2 dz dt + lnx dt of the Einstein gravitational field equation follows from the general ansatz ds = dx + gαβ(x) dx dx but fails to follow from it if the symmetric matrix gαβ(x) is assumed to be in diagonal form. KEY: Vacuum solution, Einstein field equation, symmetries, diagonalization

It is well known that a system of power polynomial equations can be reduced to a single-variable polynomial equation by exploiting the so-called Newton’s identities. In this work, by further exploring Newton’s identities, we discover a binomial decomposition rule for composite elementary symmetric polynomials. Utilizing this decomposition rule, we solve three types of systems of composite power...

This paper proposes a new representation for the predictor estimates recursion and corresponding Riccati equation for discrete-time, time-variant descriptor systems. The introduced “9-block” form for the predictor and Riccati equation presents an interesting simple and symmetric structure, which enable us to treat directly the most general systems where state and measurement noises are correlat...

We consider the nonlinear wave equation i∂tu = √ −∆+m2u− (|x| ∗ |u|)u on R modelling the dynamics of (pseudo-relativistic) boson stars. For spherically symmetric initial data, u0(x) ∈ C∞ c (R), with negative energy, we prove blow-up of u(t, x) inH-norm within a finite time. Physically, this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blow-up in exte...

In this paper, we find the exact solutions of some nonlinear evolution equations by using ( ′ G )expansion method. Four nonlinear models of physical significance i.e. the symmetric regularized longwave equation, the Klein-Gordon-Zakharov equations, the Burgers-Kadomtsev-Petviashvili equation and the nonlinear Schrödinger equation with a cubic nonlinearity are considered and obtained their exact...

A new numerical procedure is proposed to solve the symmetric matrix polynomial equation A T (?s)X(s) + X T (?s)A(s) = 2B(s) that is frequently encountered in control and signal processing. It is based on interpolation and takes fully advantage of symmetry of the equation by reducing the original problem dimension. The algorithm is more eecient and more general than older methods and, namely, it...