Dispersionless equation explained
Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see e.g. references below). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.
Examples
Dispersionless KP equation
The dispersionless Kadomtsev–Petviashvili equation (dKPE), also known (up to an inessential linear change of variables) as the Khokhlov–Zabolotskaya equation, has the form
It arises from the commutation
of the following pair of 1-parameter families of vector fields
L1=\partialy+λ\partialx-ux\partialλ, (3a)
L2=\partial
ux+uy)\partialλ, (3b)
where
is a spectral parameter. The dKPE is the
-dispersionless limit of the celebrated
Kadomtsev–Petviashvili equation, arising when considering long waves of that system. The dKPE, like many other (2+1)-dimensional integrable dispersionless systems, admits a (3+1)-dimensional generalization.
[1] The Benney moment equations
The dispersionless KP system is closely related to the Benney moment hierarchy, each of which is a dispersionless integrable system:
These arise as the consistency condition between
and the simplest two evolutions in the hierarchy are:
The dKP is recovered on setting
and eliminating the other moments, as well as identifying
and
.
If one sets
, so that the countably many moments
are expressed in terms of just two functions, the classical
shallow water equations result:
These may also be derived from considering slowly modulated wave train solutions of the
nonlinear Schrödinger equation. Such 'reductions', expressing the moments in terms of finitely many dependent variables, are described by the
Gibbons-Tsarev equation.
Dispersionless Korteweg–de Vries equation
The dispersionless Korteweg–de Vries equation (dKdVE) reads as
It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation.It is satisfied by
-independent solutions of the dKP system.It is also obtainable from the
-flow of the Benney hierarchy on setting
Dispersionless Novikov–Veselov equation
The dispersionless Novikov-Veselov equation is most commonly written as the following equation for a real-valued function
:
\begin{align}
&\partialv=\partial(vw)+\partial(v\barw),\\
&\partialw=-3\partialv,
\end{align}
where the following standard notation of complex analysis is used:
,
. The function
here is an auxiliary function, defined uniquely from
up to a holomorphic summand.
Multidimensional integrable dispersionless systems
See for systems with contact Lax pairs, and e.g.,[2] [3] and references therein for other systems.
See also
References
Bibliography
- Kodama Y., Gibbons J. "Integrability of the dispersionless KP hierarchy", Nonlinear World 1, (1990).
- Zakharov V.E. "Dispersionless limit of integrable systems in 2+1 dimensions", Singular Limits of Dispersive Waves, NATO ASI series, Volume 320, 165-174, (1994).
- 10.1142/S0129055X9500030X . Integrable Hierarchies and Dispersionless Limit . 1995 . Takasaki . Kanehisa . Takebe . Takashi . 17351327 . Reviews in Mathematical Physics . 07 . 5 . 743–808 . 1995RvMaP...7..743T . hep-th/9405096 .
- 0709.4148 . 10.1088/1751-8113/40/46/F03 . Quasiclassical generalized Weierstrass representation and dispersionless DS equation . 2007 . Konopelchenko . B. G. . 18451590 . Journal of Physics A: Mathematical and Theoretical . 40 . 46 . F995–F1004 .
- Konopelchenko . B.G. . Moro . A. . 10.1111/j.0022-2526.2004.01536.x . Integrable Equations in Nonlinear Geometrical Optics . 2004 . 17611812 . Studies in Applied Mathematics . 113 . 4 . 325–352 . nlin/0403051 . 2004nlin......3051K .
- Dunajski . Maciej . 10.1088/1751-8113/41/31/315202 . An interpolating dispersionless integrable system . 2008 . 15695718 . Journal of Physics A: Mathematical and Theoretical . 41 . 31 . 315202 . 2008JPhA...41E5202D . 0804.1234 .
- Dunajski M. "Solitons, instantons and twistors", Oxford University Press, 2010.
External links
Notes and References
- 10.1007/s11005-017-1013-4. 1401.2122. New integrable (3 + 1)-dimensional systems and contact geometry . 2018 . Sergyeyev . A. . 119159629 . Letters in Mathematical Physics . 108 . 2 . 359–376 . 2018LMaPh.108..359S .
- Calderbank . David M. J. . Kruglikov . Boris . 1612.02753 . 10.1007/s00220-020-03913-y . 3 . Communications in Mathematical Physics . 4232780 . 1811–1841 . Integrability via geometry: dispersionless differential equations in three and four dimensions . 382 . 2021.
- 1410.7104 . 10.1007/s11005-015-0800-z . Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations . 2015 . Kruglikov . Boris . Morozov . Oleg . 119326497 . Letters in Mathematical Physics . 105 . 12 . 1703–1723 . 2015LMaPh.105.1703K .