Determinants and Their Applications in Mathematical Physics

240 6. Applications of Determinants in Mathematical Physics

of magnetohydrodynamicwaves in a warmplasma, ion acousticwaves,

and acousticwaves in ananharmonic lattice. Of all physically significant

nonlinear partial differential equations with known analytic solutions, the

KdV equation is one of the simplest. The KdV equation can be regarded

as a particular case of the Kadomtsev–Petviashvili (KP) equation but it is

of such fundamental importance that it has been given detailed individual

attention in this chapter.

A method for solving the KdV equation based on the GLM integral

equation was described by Gardner, Greene, Kruskal, and Miura (GGKM)

in 1967. The solution is expressed in the form

u=2Dx{K(x, x, t)},Dx=

∂

∂x

However, GGKM did not give an explicit solution of the integral equation

and the first explicit solution of the KdV equation was given by Hirota

in 1971 in terms of a determinant with well-defined elements but of arbi-

trary order. He used an independent method which can be described as

heuristic, that is, obtained by trial and error. In another pioneering pa-

per published the same year, Zakharov solved the KdV equation using the

GGKM method. Wadati and Toda also applied the GGKM method and,

in 1972, published a solution which agrees with Hirota’s.

In 1979, Satsuma showed that the solution of the KdV equation can

be expressed in terms of a Wronskian, again with well-defined elements

but of arbitrary order. In 1982, P ̈oppe transformed the KdV equation into

an integral equation and solved it by the Fredholm determinant method.

Finally, in 1983, Freeman and Nimmo solved the KdV equation directly in

Wronskian form.

6.2.6 The Kadomtsev–Petviashvili Equation

The Kadomtsev–Petviashvili (KP) equation, namely

(ut+6uux+uxxx)x+3uyy=0,

arises in a study published in 1970 of the stability of solitarywaves in

weakly dispersive media. It can be regarded as a two-dimensional gen-

eralization of the KdV equation to which it reverts ifuis independent

ofy.

The non-Wronskian solution of the KP equation was obtained from in-

verse scattering theory (Lamb, 1980) and verified in 1989 by Matsuno using

a method based on the manipulation of bordered determinants. In 1983,

Freeman and Nimmo solved the KP equation directly in Wronskian form,

and in 1988, Hirota, Ohta, and Satsuma found a solution containing a

two-way (right and left) Wronskian. Again, all determinants have well-

defined elements but are of arbitrary order. Shortly after the Matsuno

paper appeared, A. Nakamura solved the KP equation by means of four