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., -).
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
- Integrable systems
- Nonlinear Schrödinger equation
- Nonlinear systems
- Davey–Stewartson equation
- Dispersive partial differential equation
- Kadomtsev–Petviashvili equation
- Korteweg–de Vries equation
- 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
- Ishimori_system at the dispersive equations wiki