Dispersionless equation

The dispersionless Kadomtsev–Petviashvili equation (dKPE), also known (up to an inessential linear change of variables) as the Khokhlov–Zabolotskaya equation, 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, arising when considering long waves of that system. The dKPE, like many other (2+1)-dimensional integrable dispersionless systems, admits a (3+1)-dimensional generalization.[1]

The dispersionless KP system is closely related to the Benney moment hierarchy, each of which is a dispersionless integrable system:

${\displaystyle A_{t_{2}}^{n}+A_{x}^{n+1}+nA^{n-1}A_{x}^{0}=0.}$

These arise as the consistency condition between

${\displaystyle \lambda =p+\sum _{n=0}^{\infty }A^{n}/p^{n+1},}$

and the simplest two evolutions in the hierarchy are:

${\displaystyle p_{t_{2}}+pp_{x}+A_{x}^{0}=0,}$

${\displaystyle p_{t_{3}}+p^{2}p_{x}+(pA^{0}+A^{1})_{x}=0,}$

The dKP is recovered on setting

${\displaystyle u=A^{0},}$

and eliminating the other moments, as well as identifying ${\displaystyle y=t_{2}}$ and ${\displaystyle t=t_{3}}$.

If one sets ${\displaystyle A^{n}=hv^{n}}$, so that the countably many moments ${\displaystyle A^{n}}$ are expressed in terms of just two functions, the classical shallow water equations result:

${\displaystyle h_{y}+(hv)_{x}=0,}$

${\displaystyle v_{y}+vv_{x}+h_{x}=0.}$

These may also be derived from considering slowly modulated wave train solutions of the nonlinear Schrodinger equation. Such 'reductions', expressing the moments in terms of finitely many dependent variables, are described by the Gibbons-Tsarev equation.

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

${\displaystyle u_{t_{3}}=uu_{x}.\qquad (4)}$

It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation. It is satisfied by ${\displaystyle t_{2}}$-independent solutions of the dKP system. It is also obtainable from the ${\displaystyle t_{3}}$-flow of the Benney hierarchy on setting

${\displaystyle \lambda ^{2}=p^{2}+2A^{0}.}$

The dispersionless Novikov-Veselov equation is most commonly written as the following equation for a real-valued 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 uniquely from ${\displaystyle v}$ up to a holomorphic summand.

See [1] for systems with contact Lax pairs, and e.g.,[2][3] and references therein for other systems.