The Shabat equation f 0 (t) + q 2 f 0 (qt) + f 2 (t) ? q 2 f 2 (qt) = ; t 2 R; where and q are parameters, is the simplest self-similar reduction of the so called dressing chain for constructing and analysing exactly solvable Schrr odinger equations. It is also relevant to the study of the q-oscillator algebra in quantum mechanics. The main objective of this paper is to investigate whether the ...

We study the scaling behavior of particle densities for Lévy walks whose transition length r is coupled with the transition time t as |r|∝t^{α} with an exponent α>0. The transition-time distribution behaves as ψ(t)∝t^{-1-β} with β>0. For 1<β<2α and α≥1, particle displacements are characterized by a finite transition time and confinement to |r|<t^{α} while the marginal distribution of transition...

Suppose q(z) is a smooth function on [0,∞) whose odd order derivatives are zero at z=0. Take r = (x, y, z) and let U(r, t) be the solution of the the IBVP Utt − Uxx − Uyy − Uzz + q(z)U = 0 for r ∈ R, z ≥ 0, t ∈ R Uz(x, y, z=0, t) = δ(x, y, t), U(r, t) = 0, for t < 0 We show that q(z) may be recovered from a knowledge of U(a, b, 0, t) for t varying over an interval and fixed a, b, by reducing th...

In [1], the author studied the oscillation of forced neutral delay differential equations with positive and negative coefficients, having the following form: [x(t)− R(t)x(t − r)] + P(t)x(t − τ)− Q(t)x(t − σ) = f (t), (1) where R, P, Q ∈ C([t0,∞),R) with r > 0, τ ≥ σ ≥ 0. The readers are referred to [1] for further details. The author uses an ill representation which yields a mistake. To avoid t...