In mathematics, and in particular in the theory of

soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...

s, the Dym equation (HD) is the third-order

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

u_t = u^3u_.\,

It is often written in the equivalent form for some function v of one space variable and time :

v_t=(v^)_.\,

The Dym equation first appeared in Kruskal Martin Kruskal ''Nonlinear Wave Equations''. In Jürgen Moser, editor, Dynamical Systems, Theory and Applications, volume 38 of Lecture Notes in Physics, pages 310–354. Heidelberg. Springer. 1975. and is attributed to an unpublished paper by Harry Dym. The Dym equation represents a system in which dispersion and

nonlinearity In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

are coupled together. HD is a

completely integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

evolution equation Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...

that may be solved by means of the

inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...

. It obeys an

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group) Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ...

number of

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 over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...

s; it does not possess the Painlevé property. The Dym equation has strong links to the Korteweg–de Vries equation. C.S. Gardner, J.M. Greene, Kruskal and R.M. Miura applied ym equationto the solution of corresponding problem in Korteweg–de Vries equation. The

Lax pair In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss s ...

of the Harry Dym equation is associated with the Sturm–Liouville operator. The Liouville transformation transforms this operator isospectrally into the Schrödinger operator. Fritz Gesztesy and Karl Unterkofler, Isospectral deformations for Sturm–Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys. 31 (1992), 113–137. Thus by the inverse Liouville transformation solutions of the Korteweg–de Vries equation are transformed into solutions of the Dym equation. An explicit solution of the Dym equation, valid in a finite interval, is found by an auto- Bäcklund transform :

u(t,x) = \left - 3 \alpha \left( x + 4 \alpha^2 t \right) \right .

Notes

References

* * * * * {{DEFAULTSORT:Dym Equation Solitons Exactly solvable models Integrable systems