CONNECTION BETWEEN THE KdV EQUATION AND THE TWO - DIMENSIONAL ISING MODEL

Among the various profound but solvable models tion [41 of physics, two of the most studied are the Korteweg— d2v 1 dv 2 1 dv 1 de Vries (KdV) equation for nonlinear wave phenom—i + —(av~+ i3) + ‘yu3 +-~(2) enon [1] and the two-dimensional Ising model [2] for statistical mechanics. It has been known [3] for severwherey is the independent variable and a, j3, y, and 6 al years that a Painlevé transcendent [4] of the third are constants. In fact, if one sets kind plays an important role in the two-dimensional Ising model. Recently Ablowitz and Segur [5] have a = ~ 3_i~~ shown there exists an important relation between the 2e6 266 €~ ‘ ~ 4~6 ‘ ~ = — 4~6 (3) long time asymptotic solution of the modified KdV y = I + e 2z, and v = 1 + 2ew equation (andhence indirectly the KdV equation) and a certain Painlevé transcendent of the second kind. It in (2), then (1) is obtained in the limit € —~0. is the purpose to show here that this transcendent of The thirdPainlevé equation was first introduced the second kind may be derived from the previously into the physics literature by Myers [6], where a par. studied transcendent of the third kind. Thus there is ticular case of (2) arose in the investigation of scatteran unexpected connection between these two models. ing from a finite strip. Recently the present authors The most general Painlevé equation of the second [7] explicitly constructed and analyzed the one-paramkind is [4] eter family of Painlevé transcendents of the third kind