“Dispersionless equation”的意思、由来-开放百科全书

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Dispersionless equation

释义

Examples
Dispersionless KP equation The Benney moment equations Dispersionless Korteweg–de Vries equation Dispersionless Novikov–Veselov equation Multidimensional integrable dispersionless systems
See also
References
External links

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

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, see.^[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 Schrodinger 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 :

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 ^[1] for systems with contact Lax pairs, and e.g.,^[2]^[3] and references therein for other systems.

References

1. ^¹Sergyeyev A. "New integrable (3+1)-dimensional systems and contact geometry", Lett. Math. Phys. 108 (2018), no. 2, 359-376, ArXiv:1401.2122, https://arxiv.org/abs/1401.2122 {{doi|10.1007/s11005-017-1013-4}}
2. ^Calderbank D.M.J., Kruglikov B. Integrability via geometry: dispersionless differential equations in three and four dimensions, ArXiv:1612.0275, https://arxiv.org/abs/1612.02753
3. ^Kruglikov B., Morozov O. Integrable dispersionless PDEs in 4D, their symmetry pseudogroups and deformations. Lett. Math. Phys. 105 (2015), no. 12, 1703-1723, arXiv:1410.7104, https://arxiv.org/abs/1410.7104

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).
Takasaki K., Takebe T. "Integrable Hierarchies and Dispersionless Limit", Rev. Math. Phys., 7, 743 (1995), ArXiv:hep-th/9405096, https://arxiv.org/abs/hep-th/9405096
Konopelchenko B.G. "Quasiclassical generalized Weierstrass representation and dispersionless DS equation", ArXiv: 0709.4148, https://arxiv.org/abs/0709.4148
Konopelchenko B.G., Moro A. "Integrable Equations in Nonlinear Geometrical Optics", Studies in Applied Mathematics, 113(4), pp. 325–352 (2004)
Dunajski M. "Interpolating integrable system". {{arXiv|0804.1234}}, https://arxiv.org/abs/0804.1234
Dunajski M. "Solitons, instantons and twistors", Oxford University Press, 2010.
Sergyeyev A. "New integrable (3+1)-dimensional systems and contact geometry", Lett. Math. Phys. 108 (2018), no. 2, 359-376, ArXiv:1401.2122, https://arxiv.org/abs/1401.2122
Takebe T. "Lectures on Dispersionless Integrable Hierarchies", 2014,

https://rikkyo.repo.nii.ac.jp/index.php?action=pages_view_main&active_action=repository_action_common_download&item_id=9046&item_no=1&attribute_id=22&file_no=1&page_id=13&block_id=49

External links

[https://web.archive.org/web/20080820111240/http://tosio.math.toronto.edu/wiki/index.php/Ishimori_system Ishimori_system] at the dispersive equations wiki

2 : Partial differential equations|Integrable systems

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