Drawing upon deep intuition, Ramanujan created new concepts in the theory of numbers, elliptic functions
and infinite series.
Following this discovery we came across papers by Khare and Saxena [18-20] who showed that there exist classes of nonlinear equations which possess exact solutions in the form of hyperbolic or Jacobi elliptic functions
. They showed also that besides the usual solitonic ([sech.sup.2]) and periodic cnoidal ([cn.sup.2]) solutions other new solutions in the form of superpositions of hyperbolic or Jacobi elliptic functions
Weierstrass elliptic functions
p(z) have the following addition formula:
Bowman, Introduction to Elliptic Functions
with Applications, English University Press, London, UK, 1953.
Miller, "Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions
for nonlinear PDEs," Journal of Symbolic Computation, vol.
Lang, Elliptic functions
, 2nd ed., Graduate Texts in Mathematics, 112, Springer, New York, 1987.
The famous Villat-Dini formula [12,31] (see ) solves the Dirichlet problem for a circular annulus [R.sup.-1] < [absolute value of w] < R (R > 1) in terms of the Poisson type integral with the kernel expressed through the elliptic functions
. An alternative formula via the Fourier series was also known.
To simplify the dynamic modelling the mass is concentrated at the center of the beam and exact solution is given in terms of elliptic functions
. Dynamic formulation for sliding beams that are deployed or retrieved through prismatic joint are presented by Vu-Quoc, Li .
In 2011, Kudryashov  got the general solutions of (7) via Jacobi elliptic functions
and analyzed the application of the tanh-coth method for finding exact solutions of (7) and showed that all the solutions which are presented by Wazwaz and Mehanna can be reduced to a single one and so on.
According to the periodical property of the Jacobi elliptic functions
, the density distribution [[absolute value of ([[psi].sub.1](x,t))].sup.2] is periodic.
Gong, "Some new exact solutions via the Jacobi elliptic functions
to the nonlinear Schrodinger equation," Acta Physica Sinica, vol.
Akhiezer, Elements of the Theory of Elliptic Functions