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]).

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[edit] Examples

[edit] Dispersionless KP equation

The dispersionless Kadomtsev–Petviashvili equation (dKPE) has the form

(u_t+uu_{x})_x+u_{yy}=0,\qquad (1)

It arises from the commutation

[L_1, L_2]=0.\qquad (2)

of the following pair of 1-parameter families of vector fields

L_1=\partial_y+\lambda\partial_x-u_x\partial_{\lambda},\qquad (3a)
L_2=\partial_t+(\lambda^2+u)\partial_x+(-\lambda u_x+u_y)\partial_{\lambda},\qquad (3b)

where λ is a spectral parameter. The dKPE is the x-dispersionless limit of the celebrated Kadomtsev–Petviashvili equation.

[edit] Dispersionless Korteweg–de Vries equation

The dispersionless Korteweg–de Vries equation (dKdVE) reads as

u_t=\frac{3}{2}uu_{x}.\qquad (4)

It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation.

[edit] Dispersionless Davey–Stewartson equation

[edit] Dispersionless Novikov–Veselov equation

[edit] Dispersionless Hirota equation

[edit] See also

[edit] 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
  • Dunajski M. "Interpolating integrable system". ArXiv: 0804.1234

[edit] External links