# Dispersionless equation

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

- Integrable systems
- Nonlinear Schrödinger equation
- Nonlinear systems
- Davey–Stewartson equation
- Dispersive partial differential equation
- Kadomtsev–Petviashvili equation
- Korteweg–de Vries equation

## 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

- Ishimori_system at the dispersive equations wiki