Dispersionless equation#Dispersionless KP 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

:$(u\_t+uu\_\{x\})\_x+u\_\{yy\}=0,\backslash qquad\; (1)$

It arises from the commutation

:$[L\_1,\; L\_2]=0.\backslash qquad\; (2)$

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

:$L\_1=\backslash partial\_y+\backslash lambda\backslash partial\_x-u\_x\backslash partial\_\{\backslash lambda\},\backslash qquad\; (3a)$

:$L\_2=\backslash partial\_t+(\backslash lambda^2+u)\backslash partial\_x+(-\backslash lambda\; u\_x+u\_y)\backslash partial\_\{\backslash lambda\},\backslash qquad\; (3b)$

where $\backslash lambda$ is a spectral parameter. The dKPE is the $x$-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.{{cite journal |doi=10.1007/s11005-017-1013-4|arxiv=1401.2122|title=New integrable (3 + 1)-dimensional systems and contact geometry |year=2018 |last1=Sergyeyev |first1=A. |s2cid=119159629 |journal=Letters in Mathematical Physics |volume=108 |issue=2 |pages=359–376 |bibcode=2018LMaPh.108..359S }}

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

:$A^n\_\{t\_2\}\; +\; A^\{n+1\}\_x\; +\; n\; A^\{n-1\}\; A^0\_x\; =0.$

These arise as the consistency condition between

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

and the simplest two evolutions in the hierarchy are:

:$p\_\{t\_2\}\; +\; p\; p\_x\; +\; A^0\_x\; =0,$

:$p\_\{t\_3\}\; +\; p^2\; p\_x\; +\; (p\; A^0+A^1)\_x\; =\; 0,$

The dKP is recovered on setting

:$u\; =\; A^0,$

and eliminating the other moments, as well as identifying $y=t\_2$ and $t=\; t\_3$.

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

:$h\_y\; +\; (hv)\_x=0,$

:$v\_y\; +v\; v\_x\; +\; h\_x=0.$

These may also be derived from considering slowly modulated wave train solutions of the nonlinear Schrödinger 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

:$u\_\{t\_3\}=uu\_\{x\}.\backslash qquad\; (4)$

It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation.

It is satisfied by $t\_2$-independent solutions of the dKP system.

It is also obtainable from the $t\_3$-flow of the Benney hierarchy on setting

:$\backslash lambda^2\; =\; p^2\; +\; 2A^0.$

The dispersionless Novikov-Veselov equation is most commonly written as the following equation for a real-valued function $v=v(x\_1,x\_2,t)$:

:$$

\begin{align}

& \partial_{ t } v = \partial_{ z }( v w ) + \partial_{ \bar z }( v \bar w ), \\

& \partial_{ \bar z } w = - 3 \partial_{ z } v,

\end{align}

where the following standard notation of complex analysis is used: $\backslash partial\_\{\; z\; \}\; =\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; (\; \backslash partial\_\{\; x\_1\; \}\; -\; i\; \backslash partial\_\{\; x\_2\; \}\; )$, $\backslash partial\_\{\; \backslash bar\; z\; \}\; =\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; (\; \backslash partial\_\{\; x\_1\; \}\; +\; i\; \backslash partial\_\{\; x\_2\; \}\; )$. The function $w$ here is an auxiliary function, defined uniquely from $v$ up to a holomorphic summand.

See for systems with contact Lax pairs, and e.g.,{{cite journal

| last1 = Calderbank | first1 = David M. J.

| last2 = Kruglikov | first2 = Boris

| arxiv = 1612.02753

| doi = 10.1007/s00220-020-03913-y

| issue = 3

| journal = Communications in Mathematical Physics

| mr = 4232780

| pages = 1811–1841

| title = Integrability via geometry: dispersionless differential equations in three and four dimensions

| volume = 382

| year = 2021}}{{cite journal |arxiv=1410.7104 |doi=10.1007/s11005-015-0800-z |title=Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations |year=2015 |last1=Kruglikov |first1=Boris |last2=Morozov |first2=Oleg |s2cid=119326497 |journal=Letters in Mathematical Physics |volume=105 |issue=12 |pages=1703–1723 |bibcode=2015LMaPh.105.1703K }} and references therein for other systems.

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

{{Reflist}}

• 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).
• {{cite journal |doi=10.1142/S0129055X9500030X |title=Integrable Hierarchies and Dispersionless Limit |year=1995 |last1=Takasaki |first1=Kanehisa |last2=Takebe |first2=Takashi |s2cid=17351327 |journal=Reviews in Mathematical Physics |volume=07 |issue=5 |pages=743–808 |bibcode=1995RvMaP...7..743T |arxiv=hep-th/9405096 }}
• {{cite journal |arxiv=0709.4148 |doi=10.1088/1751-8113/40/46/F03 |title=Quasiclassical generalized Weierstrass representation and dispersionless DS equation |year=2007 |last1=Konopelchenko |first1=B. G. |s2cid=18451590 |journal=Journal of Physics A: Mathematical and Theoretical |volume=40 |issue=46 |pages=F995–F1004 }}
• {{cite journal |last1=Konopelchenko |first1=B.G. |last2=Moro |first2=A. |doi=10.1111/j.0022-2526.2004.01536.x |title=Integrable Equations in Nonlinear Geometrical Optics |year=2004 |s2cid=17611812 |journal=Studies in Applied Mathematics |volume=113 |issue=4 |pages=325–352 |arxiv=nlin/0403051 |bibcode=2004nlin......3051K }}
• {{cite journal |last1=Dunajski |first1=Maciej |doi=10.1088/1751-8113/41/31/315202 |title=An interpolating dispersionless integrable system |year=2008 |s2cid=15695718 |journal=Journal of Physics A: Mathematical and Theoretical |volume=41 |issue=31 |page=315202 |bibcode=2008JPhA...41E5202D |arxiv=0804.1234 }}
• Dunajski M. "Solitons, instantons and twistors", Oxford University Press, 2010.

• [https://web.archive.org/web/20080820111240/http://tosio.math.toronto.edu/wiki/index.php/Ishimori_system Ishimori_system] at the dispersive equations wiki
• [https://rikkyo.repo.nii.ac.jp/index.php?action=pages_view_main&active_action=repository_action_common_download&item_id=9046&item_no=1&attribute_id=22&file_no=1&page_id=13&block_id=49 Takebe T. "Lectures on Dispersionless Integrable Hierarchies"], 2014

Category:Partial differential equations

Category:Integrable systems