Equation for the background [in my PDEs] If u[t, x] is independent of x , as it is sufficiently far away from the main pattern, then the partial differential equation on page 165 reduces to the ordinary differential equation u''[t]  (1 - u[t] 2 )(1 + a u[t]) u[0]  u'[0]  0 For a = 0 , the solution to this equation can be written in terms of Jacobi elliptic functions as ( √ 3 JacobiSN[t/3 1/4 , 1/2] 2 ) / (1 + JacobiCN[t/3 1/4 , 1/2] 2 ) In general the solution is (b d JacobiSN[r t, s] 2 )/(b - d JacobiCN[r t, s] 2 ) where r = -Sqrt[1/8 a c (b - d)] s = (d (c - b))/(c (d - b)) and b , c , d are determined by the equation (x - b)(x - c)(x - d)  -(12 + 6 a x - 4 x 2 - 3 a x 3 )/(3a) In all cases (except when -8/3 < a < -1/ √ 6 ), the solution is periodic and non-singular. For a = 0 , the period is 2 3 1/4 EllipticK[1/2] ≃ 4.88 . … For a = 8/3 , the solution can be written without Jacobi elliptic functions, and is given by 3 Sin[Sqrt[5/6] t] 2 /(2 + 3 Cos[Sqrt[5/6] t] 2 )

[History of] exact solutions Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) ( Sqrt ); cubic, quartic equations (1530s) ( x 1/n ); 2-body problem (1687) ( Cos ); catenary (1690) ( Cosh ); brachistochrone (1696) ( Sin ); spinning top (1849; 1888; 1888) ( JacobiSN ; WeierstrassP ; hyperelliptic functions); quintic equations (1858) ( EllipticTheta ); half-plane diffraction (1896) ( FresnelC ); Mie scattering (1908) ( BesselJ , BesselY , LegendreP ); Einstein equations (Schwarzschild (1916), Reissner–Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) ( LaguerreL , HermiteH ); 2D Ising model (1944) ( Sinh , EllipticK ); various Feynman diagrams (1960s-1980s) ( PolyLog ); KdV equation (1967) ( Sech etc.); Toda lattice (1967) ( Sech ); six-vertex spin model (1967) ( Sinh integrals); Calogero–Moser model (1971) ( Hypergeometric1F1 ); Yang–Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) ( EllipticTheta ); additive cellular automata (1984) ( MultiplicativeOrder ); Seiberg–Witten supersymmetric theory (1994) ( Hypergeometric2F1 ).

(This can be done by repeatedly making use of functional relations such as Exp[2x]  Exp[x] 2 which express f[2x] as a polynomial in f[x] ; such an approach is known to work for elementary, elliptic, modular and other functions associated with ArithmeticGeometricMean and for example DedekindEta .) … (Examples of more difficult cases include HypergeometricPFQ[a, b, 1] and StieltjesGamma[k] , where logarithmic series can require an exponential number of terms.

But for smaller e[s] one can show that Abs[m[s]]  (1 - Sinh[2 β ] -4 ) 1/8 where β can be deduced from e[s]  -(Coth[2 β ](1 + 2 EllipticK[4 Sech[2 β ] 2 Tanh[2 β ] 2 ] (-1 + 2 Tanh[2 β ] 2 )/ π )) This implies that just below the critical point e 0 = - √ 2 (which corresponds to β = Log[1 + √ 2 ]/2 ) Abs[m] ~ (e 0 - e) 1/8 , where here 1/8 is a so-called critical exponent.

(The proof is based on having bounds for how close to zero Sum[ α i , Log[ α i ], i, j] can be for independent algebraic numbers α k .) … Extensive work has been done since the early 1900s on so-called elliptic curve equations such as x 2  a y 3 + b whose corresponding algebraic surface has a single hole (genus 1).