# 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., -).

## Examples

### 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

$L_{1}=\partial _{y}+\lambda \partial _{x}-u_{x}\partial _{\lambda },\qquad (3a)$ $L_{2}=\partial _{t}+(\lambda ^{2}+u)\partial _{x}+(-\lambda u_{x}+u_{y})\partial _{\lambda },\qquad (3b)$ ### 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.

### Dispersionless Novikov–Veselov equation

The dispersionless Novikov-Veselov equation is most commonly written as the following equation on function $v=v(x_{1},x_{2},t)$ :

{\begin{aligned}&\partial _{t}v=\partial _{z}(vw)+\partial _{\bar {z}}(v{\bar {w}}),\\&\partial _{\bar {z}}w=-3\partial _{z}v,\end{aligned}} 