# 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

${\displaystyle (u_{t}+uu_{x})_{x}+u_{yy}=0,\qquad (1)}$

It arises from the commutation

${\displaystyle [L_{1},L_{2}]=0.\qquad (2)}$

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

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

where ${\displaystyle \lambda }$ is a spectral parameter. The dKPE is the ${\displaystyle x}$-dispersionless limit of the celebrated Kadomtsev–Petviashvili equation.

### Dispersionless Korteweg–de Vries equation

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

${\displaystyle u_{t}={\frac {3}{2}}uu_{x}.\qquad (4)}$

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 ${\displaystyle v=v(x_{1},x_{2},t)}$:

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

where the following standard notation of complex analysis is used: ${\displaystyle \partial _{z}={\frac {1}{2}}(\partial _{x_{1}}-i\partial _{x_{2}})}$, ${\displaystyle \partial _{\bar {z}}={\frac {1}{2}}(\partial _{x_{1}}+i\partial _{x_{2}})}$. The function ${\displaystyle w}$ here is an auxiliary function defined via ${\displaystyle v}$ up to a holomorphic summand. The function ${\displaystyle v}$ is generally assumed to be a real-valued function.