Dispersionless equation

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Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and are intensively studied in the recent literature (see, f.i., [1]-[5]).

Examples

Dispersionless KP equation

The dispersionless Kadomtsev–Petviashvili equation (dKPE) 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.

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.

Dispersionless Davey–Stewartson equation

Dispersionless Novikov–Veselov equation

The dispersionless Novikov-Veselov equation is most commonly written as the following equation on function :

where the following standard notation of complex analysis is used: , . The function here is an auxiliary function defined via up to a holomorphic summand. The function is generally assumed to be a real-valued function.

Dispersionless Hirota equation

See also

References

  • Kodam Y., Gibbons J. "Integrability of the dispersionless KP hierarchy"
  • Zakharov V.E. "Dispersionless limit of integrable systems in 2+1 dimensions"
  • Takasaki K., Takebe T. Rev. Math. Phys., 7, 743 (1995)
  • Konopelchenko B.G. "Quasiclassical generalized Weierstrass representation and dispersionless DS equation", ArXiv: 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

External links