Page 255 - Determinants and Their Applications in Mathematical Physics

240 6. Applications of Determinants in Mathematical Physics

of magnetohydrodynamic waves in a warm plasma, ion acoustic waves,
and acoustic waves in an anharmonic 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 =2D x {K(x, x, t)}, D x = .
∂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
(u t +6uu x + u xxx ) x +3u yy =0,
arises in a study published in 1970 of the stability of solitary waves in
weakly dispersive media. It can be regarded as a two-dimensional gen-
eralization of the KdV equation to which it reverts if u is independent
of y.
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