- Phase space method
In

applied mathematics , the**phase space method**is a technique for constructing and analyzing solutions ofdynamical system s, that is, solving time-dependentdifferential equations . The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables. The original and the new variables form a vector in thephase space . The solution then becomes acurve in the phase space, parametrized by time. The curve is usually called atrajectory or an orbit. The differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization. Finally, a solution in the phase space is transformed back into the original setting.The phase space method is used widely in

physics . It can be applied, for example, to findtraveling wave solutions ofreaction-diffusion system s. A. Kolmogorov, I. Petrovskii, and N. Piscounov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In V. M. Tikhomirov, editor, "Selected Works of A. N. Kolmogorov I", pages 248--270. Kluwer 1991. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1--25, 1937] Peter Grindrod. "The theory and applications of reaction-diffusion equations: Patterns and waves." Oxford Applied Mathematics and Computing Science Series. The Clarendon Press Oxford University Press, New York, second edition, 1996.]**ee also***

Reaction-diffusion system

*Fisher's equation

*Allen-Cahn equation **References**

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