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A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing controlled Brownian behavior of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock price and time. The model includes two parameters: volatility (playing the role of dispersion frequency coefficient), which can be either fixed or stochastic, and adaptive market potential that depends on the interest rate. The wave function represents quantum probability amplitude, whose absolute square is probability density function. Four types of analytical solutions of the NLS equation are provided in terms of Jacobi elliptic functions, all starting from de Broglie's plane-wave packet associated with the free quantum-mechanical particle. The best agreement with the Black-Scholes model shows the adaptive shock-wave NLSsolution, which can be efficiently combined with adaptive solitary-wave NLS-solution. Adjustable 'weights' of the adaptive market-heat potential are estimated using either unsupervised Hebbian learning or supervised LevenbergMarquardt algorithm. In the case of stochastic volatility, it is itself represented by the wave function, so we come to the so-called Manakov system of two coupled NLS equations (that admits closed-form solutions), with the common adaptive market potential, which defines a bidirectional spatio-temporal associative memory.

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In this paper we present rotational symplectic geometry for use in modern biomechanics of human motion. Generalized Hamiltonian formalism has been formulated on the configuration manifold consisting of rotational Lie groups. The proposed symplectic biodynamics includes conservative, dissipative, and driving forces and spinal reflex-like servo controls. Reduction of the angular configuration manifold enables topological brain-like control, as well as a subsequent topological analysis.

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