A proposed method of detecting moving targets in scenes that include cluttered or noisy backgrounds is based on a soliton-resonance mathematical model. The model is derived from asymptotic solutions of the cubic Schroedinger equation for a one-dimensional system excited by a position-and-time- dependent externally applied potential. The cubic Schroedinger equation has general significance for time-dependent dispersive waves. It has been used to approximate several phenomena in classical as well as quantum physics, including modulated beams in nonlinear optics, and superfluids (in particular, Bose-Einstein condensates). In the proposed method, one would take advantage of resonant interactions between (1) a soliton excited by the position-and-time-dependent potential associated with a moving target and (2) "eigen-solitons," which represent dispersive waves and are solutions of the cubic Schroedinger equation for a time-independent potential.

In nondimensionalized form, the cubic Schroedinger equation is

*iu _{t}* +

*u*+ v|

_{xx}*u*|

^{2}

*u*=

*Vu*,

where *x* is the nondimensionalized position coordinate, *t* is nondimensionalized time, *u(x,t)* is a complex state variable, n is a coupling constant, *V(x,t)* is the nondimensionalized externally applied potential, and the subscripts denote partial differentiation with respect to the variables shown therein. The equation admits of a variety of solutions that have different qualitative and quantitative properties: Depending on the magnitudes and signs of model parameters, the model can represent a positive or negative moving target potential that induces "bright" or "dark" solitons in an attractive or repulsive Bose-Einstein condensate.

In the proposed method, one would exploit a property of "bright" soliton solutions: Any uniformly moving component of an external potential (for example, representing uniform motion of a target) is amplified, while the remaining components (for example, representing noise) are dispersed. This phenomenon is similar to a classical resonance, in which out-of-resonance components eventually vanish.

A target-detection algorithm according to the proposed method would begin with conversion of readings of target-motion-detecting sensors into values of a fictitious moving potential. The values would, in turn, be fed as input to a computational model of a dynamic system governed by the cubic Schroedinger equation. The only surviving output signal components would be those having space and time dependence proportional to the moving potential. The algorithm could be implemented computationally — possibly by use of a neural-network mathematical model. Alternatively, the algorithm could be implemented by use of a physical model — for example, a superfluid or a nonlinear optical system.

The algorithm could be expanded to detect a moving target in a two- or three-dimensional space. It would not be necessary to develop a two- or three-dimensional soliton-resonance model. For this purpose, it would suffice to use two or three one-dimensional soliton-resonance models, each of which would be used to detect the projection of the motion of the target onto one of the two or three coordinate axes. One would then construct a representation of the two- or three-dimensional target motion from the outputs of the algorithm for the two or three axes.

*This work was done by Michael Zak and Igor Kulikov of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com/tsp under the Information Sciences category. NPO-30895. *

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###### Detecting Moving Targets by Use of Soliton Resonances

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This article first appeared in the October, 2003 issue of *NASA Tech Briefs* Magazine.

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