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Journal of Mathematical Physics / Volume 26 / Issue 1

J. Math. Phys. 26, 68 (1985); http://dx.doi.org/10.1063/1.526750 (6 pages)

Invariants for dissipative nonlinear systems by using rescaling

Marc R. Feix and H. Ralph Lewis

Los Alamos National Laboratory, Center for Nonlinear Studies and CTR Division, Los Alamos, New Mexico 87545

(Received 5 January 1984; accepted 28 June 1984)

A rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second‐order differential equation with a friction term proportional to the velocity, β(t)v. The transformation is of the form (x,t)→(ξ,θ), where x=ξC(t)+α(t), dθ=dt/A2(t). This rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant. The invariants are found explicitly for a power‐law potential, γ(t)xm+1/(m+1), and an arbitrary coefficient of friction β(t). We show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist. For certain parameters, the asymptotic solution is a self‐similar solution that is an attractor for all initial conditions. The technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications.

KEYWORDS and PACS

Keywords

transformations, space−time, differential equations, velocity, potentials, analytical solution, friction, dissipation factor, magnetohydrodynamics, numerical solution, disturbances

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.526750

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
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    H. R. Lewis and P. G. L. Leach, J. Math. Phys. 23, 2371 (1982JMAPAQ000023000012002371000001).

    H. R. Lewis, J. Math. Phys. 25, 1139 (1984JMAPAQ000025000004001139000001).

    H. R. Lewis and K. R. Symon, Phys. Fluids 27, 192 (1984PFLDAS000027000001000192000001).

    B. Abraham-Shrauner, Phys. Fluids 27, 197 (1984PFLDAS000027000001000197000001).

    A. Munier, J. R. Burgan, M. R. Feix, and E. Fijalkow, J. Math. Phys. 22, 1219 (1981JMAPAQ000022000006001219000001).

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