Exactly solvab q-extended nonlinear classical and quantum models

Abstract

In the present thesis we study q-extended exactly solvable nonlinear classical and quantum models. In these models the derivative operator is replaced by q-derivative, in the form of finite difference dilatation operator. It requires introducing q-numbers instead of standard numbers, and q-calculus instead of standard calculus. We start with classical q-damped oscillator and q-difference heat equation. Exact solutions are constructed as q-Hermite and Kampe-de Feriet polynomials and Jackson q-exponential functions. By q-Cole-Hopf transformation we obtain q-nonlinear heat equation in the form of Burgers equation. IVP for this equation is solved in operator form and q-shock soliton solutions are found. Results are extended to linear q-Schrödinger equation and nonlinear q-Maddelung fluid. Motivated by physical applications, then we introduce the multiple q-calculus. In addition to non-symmetrical and symmetrical q-calculus it includes the new Fibonacci calculus, based on Binet-Fibonacci formula. We show that multiple q-calculus naturally appears in construction of Q-commutative q-binomial formula, generalizing all well-known formulas as Newton, Gauss, and noncommutative ones. As another application we study quantum two parametric deformations of harmonic oscillator and corresponding q-deformed quantum angular momentum. A new type of q-function of two variables is introduced as q-holomorphic function, satisfying q-Cauchy-Riemann equations. In spite of that q-holomorphic function is not analytic in the usual sense, it represents the so-called generalized analytic function. The q-traveling waves as solutions of q-wave equation are derived. To solve the q-BVP we introduce q-Bernoulli numbers, and their relation with zeros of q-Sine function.