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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 = 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 )

Other standard mathematical functions that oscillate at large x include JacobiSN and MathieuC .

[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 ).

(It has long been known that only elliptic functions such as JacobiSN satisfy polynomial addition formulas—but there is no immediate analog of this for replication formulas.)