The Uncertainty Relation of One-Soliton Solution with Nonlinear Schrödinger Equation and Its Performance as an Analyzing Wavelet

Abstract

Wavelet transform (WT) of the modulating multiple-soliton solution for the nonlinear Schrödinger (NLS) equation is performed with the use of a hyperbolic function, the one-soliton solution of the NLS equation, as an analyzing wavelet (AW).The uncertainty relation of this new hyperbolic AW is explicitly obtained as Δ x Δ p = π/6. Suitability of the hyperbolic AW is also examined and its performance is compared with that due to the Gabor AW via conventional statistical mechanical quantities such as the mean value, the most probable value, the skewness and the kurtosis of position-momentum (x-p) distribution for modulating solitons.