# Dispersionless equation

Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see, f.i., [1]-[5]). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.

## 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, arising when considering long waves of that system.

### The Benney moment equations

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.

### Dispersionless Korteweg–de Vries 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

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

### Dispersionless Novikov–Veselov equation

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.

## References

• Kodama Y., Gibbons J. "Integrability of the dispersionless KP hierarchy", Nonlinear World 1, (1990).
• Zakharov V.E. "Dispersionless limit of integrable systems in 2+1 dimensions", Singular Limits of Dispersive Waves, NATO ASI series, Volume 320, 165-174, (1994).
• Takasaki K., Takebe T. "Integrable Hierarchies and Dispersionless Limit", Rev. Math. Phys., 7, 743 (1995), ArXiv:hep-th/9405096, https://arxiv.org/abs/hep-th/9405096
• Konopelchenko B.G. "Quasiclassical generalized Weierstrass representation and dispersionless DS equation", ArXiv: 0709.4148, https://arxiv.org/abs/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, https://arxiv.org/abs/0804.1234
• Dunajski M. "Solitons, instantons and twistors", Oxford University Press, 2010.
• Sergyeyev A. "New integrable (3+1)-dimensional systems and contact geometry", Lett. Math. Phys. 108 (2018), no. 2, 359-376, ArXiv:1401.2122, https://arxiv.org/abs/1401.2122
• Takebe T. "Lectures on Dispersionless Integrable Hierarchies", 2014,