KP-equation

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 [a4] but was soon recognized to be related to problems of algebraic geometry and representation theory, besides spectral theory.

The general (formal) solution was found by M. Sato, cf. [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 [a5] in terms of theta-functions on the Jacobi variety of a complex curve (cf. also Complex manifold) and T. Shiota [a11] settled the Novikov conjecture by proving that, conversely, if a theta-function satisfies the KP-equation, then its period matrix comes from a Riemann surface (cf. 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 $\overline { \partial }$-problem, cf. [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, 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. [a7].

The celebrated Korteweg–de Vries equation (respectively, the Boussinesq equation, see also Turbulence, mathematical problems in; Soliton; 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), obtained by possibly modifying the group whose representation theory yields the tau-function ($\operatorname{GL} ( \infty )$ in the KP-case), cf. [a3].

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

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

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

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

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

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

#### References[edit]

[a1] |  M.J. Ablowitz, P.A. Clarkson, "Solitons, nonlinear evolution equations and inverse scattering" , Cambridge Univ. Press (1991) MR1149378 Zbl 0762.35001

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[a2] |  R. Beals, P. Deift, C. Tomei, "Direct and inverse scattering on the line" , Amer. Math. Soc. (1988) MR0954382 Zbl 0679.34018
[a3] |  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 MR0725700 Zbl 0571.35099
[a4] |  B.B. Kadomtsev, V.J. Petviashvili, "On the stability of solitary waves in weakly dispersive media" Soviet Phys. Dokl. , 15 (1970) pp. 539–541
[a5] |  I.M. Krichever, "Methods of algebraic geometry in the theory of non-linear equations" Russian Math. Surveys , 32 : 6 (1977) pp. 185–213 Zbl 0461.35075
[a6] |  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 MR0601756 Zbl 0548.35100 Zbl 0501.35071
[a7] |  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) MR1314667 Zbl 0837.35132
[a8] |  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 MR1104527 Zbl 0732.14008
[a9] |  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 MR1355554 Zbl 0897.58022
[a10] |  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 MR1013125 Zbl 0688.58016
[a11] |  T. Shiota, "Characterization of Jacobian varieties in terms of soliton equations" Invent. Math. , 83 (1986) pp. 333–382 MR0818357 Zbl 0621.35097
[a12] |  E. Witten, "Two-dimensional gravity and intersection theory on moduli space" J. Diff. Geom. , Suppl. 1 (1991) pp. 243–310 MR1144529 Zbl 0757.53049