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The Painlevé equations were derived by Painlevé and Gambier in the years 1895 − 1910. Given a rational function R in w, w ′ and analytic in z, they searched what were the second order ordinary differential equations of t...
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Rational solutions to the Boussinesq equation are constructed as a quotient of two polynomials in x and t. For each positive integer N , the numerator is a polynomial of degree N (N + 1) − 2 in x and t, while the denomin...