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''Kadomtsev–Petviashvili equation''

The equation

\begin{equation*} ( u _ { t } + 6 u u _ { x } + u _ { xxx } ) _ { x } + 3 \sigma ^ { 2 } u _ { yy }  = 0, \end{equation*}

with $\sigma = \pm 1$. It arose in applied mathematics [[#References|[a4]]] but was soon recognized to be related to problems of [[Algebraic geometry|algebraic geometry]] and [[Representation theory|representation theory]], besides [[Spectral theory|spectral theory]].

The general (formal) solution was found by M. Sato, cf. [[#References|[a10]]], and is given as:

\begin{equation*} 2 \frac { \partial ^ { 2 } } { \partial x ^ { 2 } } \operatorname { log } \tau, \end{equation*}

by the tau-function, the section of a determinant line bundle over an infinite-dimensional Grassmannian; Sato proved that the KP-equation is equivalent to the Plücker relations for this Grassmannian.

The algebro-geometric solution was expressed by I.M. Krichever [[#References|[a5]]] in terms of theta-functions on the [[Jacobi variety|Jacobi variety]] of a complex curve (cf. also [[Complex manifold|Complex manifold]]) and T. Shiota [[#References|[a11]]] settled the Novikov conjecture by proving that, conversely, if a [[Theta-function|theta-function]] satisfies the KP-equation, then its period matrix comes from a [[Riemann surface|Riemann surface]] (cf. [[Schottky problem|Schottky problem]]).

The spectral theory, an inverse scattering in two space and one time dimensions, was attacked most effectively by R. Beals and R.R. Coifman by designing the [[Neumann d-bar problem|Neumann $\overline { \partial }$-problem]], cf. [[#References|[a2]]].

All these methods treat also the KP-hierarchy, of which the KP-equation is the first non-trivial member. This hierarchy can be formulated for functions of infinitely many variables $\mathbf t = ( t _ { j } )$ as the identity for the coefficients of

\begin{equation*} \frac { \partial } { \partial t _ { j } } \mathcal{L} = [ ( \mathcal{L} ^ { j } ) _ { + } , \mathcal{L} ], \end{equation*}

where $\mathcal{L} = \partial + u _ { - 1 } ( x ) \partial ^ { - 1 } + u _ { - 2 } ( x ) \partial ^ { - 2 } +\dots$ is a (formal) [[Pseudo-differential operator|pseudo-differential operator]], the variables of the KP-equation are taken to be $x = t_1$, $y = t_2$, $t = t _ { 3 }$, $\partial / \partial x$ is abbreviated as $\partial$ and $( - )_+$ denotes deletion of the terms involving negative powers of $\partial$. From this point of view, the curve associated to the algebro-geometric solutions is the spectral curve of the ring of differential operators in $x$ that commute with $\mathcal{L}$ and the times $t_j$ are the isospectral deformations of the problem, which are linear flows on the Jacobi variety, a complex torus $\mathbf{C} / \Lambda$; this theory was developed by J.L. Burchnall and T.W. Chaundy in the 1920s, cf. [[#References|[a7]]].

The celebrated [[Korteweg–de Vries equation|Korteweg–de Vries equation]] (respectively, the Boussinesq equation, see also [[Turbulence, mathematical problems in|Turbulence, mathematical problems in]]; [[Soliton|Soliton]]; [[Oberbeck–Boussinesq equations|Oberbeck–Boussinesq equations]]) is but a reduction of the KP-equation, in the sense that the solution $u ( x , y , t )$ is independent of $y$ (respectively, $t$), as are other soliton equations ( cf. also [[Soliton|Soliton]]), obtained by possibly modifying the group whose representation theory yields the tau-function ($\operatorname{GL} ( \infty )$ in the KP-case), cf. [[#References|[a3]]].

Numerous issues related to the KP-equation are still under active investigation:

1) reality conditions (cf. [[#References|[a9]]]), for the aforementioned solutions are given over the complex numbers;

2) solutions that belong to special classes of functions (cf. [[#References|[a1]]]), such as rational, solitonic, elliptic, bispectral;

3) construction of solutions from vector bundles of rank $r$ over a curve (cf. [[#References|[a6]]]), which generalize the Jacobian case where $r = 1$;

4) construction of analogous hierarchies for commuting matrix differential operators (cf. [[#References|[a8]]]), whose spectral variety has dimension greater than $1$; and

5) connections with matrix models and [[Quantum field theory|quantum field theory]] (cf. [[#References|[a12]]]).

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> M.J. Ablowitz, P.A. Clarkson, "Solitons, nonlinear evolution equations and inverse scattering" , Cambridge Univ. Press (1991) {{MR|1149378}} {{ZBL|0762.35001}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> R. Beals, P. Deift, C. Tomei, "Direct and inverse scattering on the line" , Amer. Math. Soc. (1988) {{MR|0954382}} {{ZBL|0679.34018}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> E. Date, M. Kashiwara, M. Jimbo, T. Miwa, "Transformation groups for soliton equation" , ''Nonlinear Integrable Systems - Classical Theory and Quantum Theory Proc. RIMS Symp., Kyoto 1981'' (1983) pp. 39–119 {{MR|0725700}} {{ZBL|0571.35099}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> B.B. Kadomtsev, V.J. Petviashvili, "On the stability of solitary waves in weakly dispersive media" ''Soviet Phys. Dokl.'' , '''15''' (1970) pp. 539–541</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> I.M. Krichever, "Methods of algebraic geometry in the theory of non-linear equations" ''Russian Math. Surveys'' , '''32''' : 6 (1977) pp. 185–213 {{MR|}} {{ZBL|0461.35075}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> I.M. Krichever, S.P. Novikov, "Holomorphic bundles over algebraic curves and nonlinear equations" ''Russian Math. Surveys'' , '''35''' : 6 (1980) pp. 53–79 ''Uspekhi Mat. Nauk'' , '''35''' : 6 (216) (1980) pp. 47–68 {{MR|0601756}} {{ZBL|0548.35100}} {{ZBL|0501.35071}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> M. Mulase, "Algebraic theory of the KP equations" R. Penner (ed.) et al. (ed.) , ''Perspectives in Mathematical Physics. Proc. Conf. Interface Math. And Physics, Taiwan summer 1992'' , ''Conf. Proc. Math. Phys.'' , '''3''' , Internat. Press (1994) pp. 151–217 (Also: Special Session On Topics In Geometry And Physics, Los Angeles, Winter 1992) {{MR|1314667}} {{ZBL|0837.35132}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> A. Nakayashiki, "Structure of Baker–Akhiezer modules of principally polarized abelian varieties, commuting partial differential operators and associated integrable systems" ''Duke Math. J.'' , '''62''' : 2 (1991) pp. 315–358 {{MR|1104527}} {{ZBL|0732.14008}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> S.M. Natanzon, "Real nonsingular finite zone solutions of soliton equations" S.P. Novikov (ed.) , ''Topics in Topol. Math. Physics'' , ''Transl. Ser. 2'' , '''170''' , Amer. Math. Soc. (1995) pp. 153–183 {{MR|1355554}} {{ZBL|0897.58022}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> M. Sato, "The KP hierarchy and infinite-dimensional Grassmann manifolds" , ''Theta functions. Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987'' , ''Proc. Symp. Pure Math.'' , '''49:1''' , Amer. Math. Soc. (1989) pp. 51–66 {{MR|1013125}} {{ZBL|0688.58016}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> T. Shiota, "Characterization of Jacobian varieties in terms of soliton equations" ''Invent. Math.'' , '''83''' (1986) pp. 333–382 {{MR|0818357}} {{ZBL|0621.35097}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> E. Witten, "Two-dimensional gravity and intersection theory on moduli space" ''J. Diff. Geom.'' , '''Suppl. 1''' (1991) pp. 243–310 {{MR|1144529}} {{ZBL|0757.53049}} </td></tr></table>