Dym equation - Citizendia

In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation

$u_t = u^3u_{xxx}.\,$

It is often written in the equivalent form

$v_t=(v^{-1/2})_{xxx}.\,$

The Dym equation first appeared in Kruskal ^[1] and is attributed to an unpublished paper by Harry Dym. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics and Physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse that maintains its shape while it travels at constant speed In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Professor Harry Dym (הארי דיםis a mathematician at the Weizmann Institute of Science, Israel.

The Dym equation represents a system in which dispersion and nonlinearity are coupled together. This article describes the use of the term nonlinearity in mathematics HD is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. In Mathematics and Physics, there are various distinct notions that are referred to under the name of integrable systems. This article describes the use of the term nonlinearity in mathematics In Mathematics, the inverse scattering transform is a method for solving some non-linear Partial differential equations. It is interesting because it obeys an infinite number of conservation laws; it does not possess the Painlevé property. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In mathematics Painlevé transcendents are solutions to certain Nonlinear second-order ordinary differential equations in the complex plane with the

The Dym equation has strong links to the Korteweg–de Vries equation. In Mathematics, the Korteweg–de Vries equation ( KdV equation for short is a Mathematical model of waves on shallow water surfaces The Lax pair of the Harry Dym equation is associated with the Sturm-Liouville operator. In Mathematics, in the theory of Differential equations, a Lax pair is a pair of time-dependent matrices that describe certain solutions of differential equations In Mathematics and its applications a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855 and Joseph Liouville The Liouville transformation transforms this operator isospectrally into the Schrödinger operator. ^[2]

Notes

^ Kruskal, M. Nonlinear Wave Equations. Martin David Kruskal ( September 28 1925 &ndash December 26 2006) was an American Mathematician and Physicist. In J. Moser, editor, Dynamical Systems, Theory and Applications, volume 38 of Lecture Notes in Physics, pages 310-354. Heidelberg. Springer. 1975.
^ F. Gesztesy and K. Unterkofler, Isospectral deformations for Sturm-Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys. 31 (1992), 113-137.

References

Cercignani, Carlo; David H. Sattinger (1998). Scaling limits and models in physical processes. Basel: Birkhäuser Verlag. ISBN 0817659854.

Kichenassamy, Satyanad (1996). Nonlinear wave equations. Marcel Dekker. ISBN 0824793285.

Gesztesy, Fritz; Holden, Helge (2003). Soliton equations and their algebro-geometric solutions. Cambridge University Press. ISBN 0521753074.

Olver, Peter J. (1993). Applications of Lie groups to differential equations, 2nd ed. Springer-Verlag. ISBN 0387940073.

This mathematical analysis-related article is a stub. Analysis has its beginnings in the rigorous formulation of Calculus. You can help Wikipedia by expanding it.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |