Negative Binomial States of the Radiation Field and their Excitations are Nonlinear Coherent States

We show that the well-known negative binomial states of the radiation field and their excitations are nonlinear coherent states. Excited nonlinear coherent state are still nonlinear coherent states with different nonlinear functions. We finally give exponential form of the nonlinear coherent states and remark that the binomial states are not nonlinear coherent states.Comment: 10 pages, no figure

MIMO nonlinear PID predictive controller

A class of nonlinear generalised predictive controllers (NGPC) is derived for multi-input multi-output (MIMO) nonlinear systems with offset or steady-state response error. The MIMO composite controller consists of an optimal NGPC and a nonlinear disturbance observer. The design of the nonlinear disturbance observer to estimate the offset is particularly simple, as is the associated proof of overall nonlinear closed-loop system stability. Moreover, the transient error response of the disturbance observer can be arbitrarily specified by simple design parameters. Very satisfactory performance of the proposed MIMO nonlinear predictive controller is demonstrated for a three-link nonlinear robotic manipulator example

Nonlinear guided waves and spatial solitons in a periodic layered medium

We overview the properties of nonlinear guided waves and (bright and dark) spatial optical solitons in a periodic medium created by a sequence of linear and nonlinear layers. First, we consider a single layer with a cubic nonlinear response (a nonlinear waveguide) embedded into a periodic layered linear medium, and describe nonlinear localized modes (guided waves and Bragg-like localized gap modes) and their stability. Then, we study modulational instability as well as the existence and stability of discrete spatial solitons in a periodic array of identical nonlinear layers, a one-dimensional nonlinear photonic crystal. Both similarities and differences with the models described by the discrete nonlinear Schrodinger equation (derived in the tight-binding approximation) and coupled-mode theory (valid for the shallow periodic modulations) are emphasized.Comment: 10 pages, 14 figure

Nonlinear cointegration and nonlinear error correction

The relationships between stochastic trending variables given by the concepts of cointegration and error correction (EC) are well characterized in a linear context, but the extension to a nonlinear context is still a challenge. Few extensions of the linear framework were developed in the context of linear cointegration but nonlinear error correction (NEC) models, and even in this context, there are still many open questions. The theoretical framework is not well developed at this moment and only particular cases have been discussed empirically. In this paper we propose a statistical framework that allow us to address those issues. First, we generalize the notion of integration to the nonlinear case. As a result a generalization of cointegration is feasible, and also a formal definition of NEC models. Within this framework we analyze the nonlinear least squares (NLS) estimation of nonlinear cointegration relations and the extension of the two-step estimation procedures of Engle and Granger (1987) for NEC models. Finally, we discuss a generalization of Granger Representation Theorem to the nonlinear case and discuss the properties of the onestep (NLS) procedure to estimate NEC models

Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations

For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions \cn(x,m) and \dn(x,m) with modulus $m$, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schr\"odinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schr\"odinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schr\"odinger equation, $\lambda \phi^4$, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of \dn^2(x,m), it also admits solutions in terms of \dn^2(x,m) \pm \sqrt{m} \cn(x,m) \dn(x,m), even though \cn(x,m) \dn(x,m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.Comment: 40 pages, no figure

Two-mode Nonlinear Coherent States

Two-mode nonlinear coherent states are introduced in this paper. The pair coherent states and the two-mode Perelomov coherent states are special cases of the two-mode nonlinear coherent states. The exponential form of the two-mode nonlinear coherent states is given. The photon-added or photon-subtracted two-mode nonlinear coherent states are found to be two-mode nonlinear coherent states with different nonlinear functions. The parity coherent states are introduced as examples of two-mode nonlinear coherent states, and they are superpositions of two corresponding coherent states. We also discuss how to generate the parity coherent states in the Kerr medium.Comment: 11 pages, no figures, accepted for publication in Optics Communication