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Solid-State 104, Springer-Verlag, Berlin, 1993. , Ginsparg, P. : 2D gravity and random matrices, Phys. Rep. 254 (1995), 1–133. : Monotone Matrix Function and Analytic Continuation, Grundlehren Math. Wiss. 207, Springer-Verlag, New York, 1974. : An Introduction to Statistical Analysis of Random Arrays, VSP, Utrecht, 1998. Hertz, J. , Krogh, A. and Palmer, R. : Introduction to the Theory of Neural Computations, Santa Fe Inst. Stud. Sci. Complexity Lecture Notes, I, Addison-Wesley, Redwood City, CA, 1991.

Any two-soliton solution is expressible as [1] u(x, t) = 2∂x2 ln τ (x, t). 5) It follows from this formula that the poles of u(x, t) are the zeros of τ (x, t) if τ (x, t) is entire in x. Then the Weierstrass Factorization Theorem [5] gives a factorization for τ (x, t): ∞ 1− τ (x, t) = C k=1 x x/xk e . 6) Since only the second logarithmic derivative of this function is relevant, the constant C is not important. 7) where the first product runs over the poles in the fundamental strip. The second product runs over all strips.

11a) x˙1 = −4k22 , corresponding to the dynamics of the one-soliton solution. 4a) that the velocity of these vertical lines is given by the one-soliton velocity for 58 BERNARD DECONINCK AND HARVEY SEGUR each line, as expected. 12) valid for any integer p > 1. • A full analysis of the interaction of the poles for the case of any two-soliton solution with k1 /k2 = p/q is given in [17]. 3. 2) m∈Z g a hyperelliptic Riemann theta function of genus g. , symmetric and negative definite) B originates from a hyperelliptic Riemann surface with only one point at infinity.