Dym equation
Encyclopedia
In mathematics
, and in particular in the theory of soliton
s, the Dym equation (HD) is the third-order partial differential equation
It is often written in the equivalent form
The Dym equation first appeared in Kruskal and is attributed to an unpublished paper by Harry Dym
.
The Dym equation represents a system in which dispersion
and nonlinearity
are coupled together. HD is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform
. It is interesting because it obeys an infinite number of conservation law
s; it does not possess the Painlevé property.
The Dym equation has strong links to the Korteweg–de Vries equation
. The Lax pair
of the Harry Dym equation is associated with the Sturm–Liouville operator.
The Liouville transformation transforms this operator isospectrally into the Schrödinger operator.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and in particular in the theory of soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
s, the Dym equation (HD) is the third-order partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
It is often written in the equivalent form
The Dym equation first appeared in Kruskal and is attributed to an unpublished paper by Harry Dym
Harry Dym
Professor Harry Dym is a mathematician at the Weizmann Institute of Science, Israel. Dym's research interests include operator theory, interpolation theory, and inverse problems.He is notable for discovering the Dym equation.-External links:...
.
The Dym equation represents a system in which dispersion
Dispersion
Dispersion may refer to:In physics:*The dependence of wave velocity on frequency or wavelength:**Dispersion , for light waves**Dispersion **Acoustic dispersion, for sound waves...
and nonlinearity
Nonlinearity
In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...
are coupled together. HD is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform
Inverse scattering transform
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years...
. It is interesting because it obeys an infinite number of conservation law
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
s; it does not possess the Painlevé property.
The Dym equation has strong links to the Korteweg–de Vries equation
Korteweg–de Vries equation
In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...
. The Lax pair
Lax pair
In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that describe the corresponding differential equations. They were introduced by Peter Lax to discuss solitons in continuous media...
of the Harry Dym equation is associated with the Sturm–Liouville operator.
The Liouville transformation transforms this operator isospectrally into the Schrödinger operator.