As we have previously pointed out, Elias-Zuniga concludes that "since the Jacobian elliptic function cn(toi, m) has a period in tot equal to 4K(m), we may see that the corresponding exact period of oscillation T is given by T = 4K(m)/[omega]" [11, page 2576].
Milne-Thomson, "Jacobian elliptic functions and theta functions," in Handbook of Mathematical Functions with Formulas, Graphics and Mathematical Tables, M.
We have the hyperbolic function solutions, trigonometric function solutions, and the Jacobian elliptic function solutions.
In this section, corresponding to all phase orbits given by Section 2, through qualitative analysis and the Jacobian elliptic functions , we discuss the exact travelling wave solutions of (3).
As a result, more types of exact solutions to (1) are obtained, which include solitons, kink solutions, and Jacobian elliptic function
solutions with double periods.
Then, we get some new periodic wave solutions in parameter forms of Jacobian elliptic function
, and numerical simulation verifies the validity of these periodic solutions.
we obtain the following relation between the Weierstrass function and the Jacobian elliptic function
: p(z, [g.sub.2], [g.sub.3]) = [e.sub.3] + ([e.sub.1] - [e.sub.3])[ns.sup.2] (z[square root of [e.sub.1] - [e.sub.3])]| M).
occur in the theory of Jacobian theta functions and their applications to Jacobian elliptic functions
. For example, Eq.
Here the modulus k of Jacobian elliptic functions
is determined uniquely as [k.sup.2] = [[A.sup.2] [b.sup.2] - [B.sup.2] [a.sup.2]]/[[b.sup.2] ([A.sup.2] - [a.sup.2])] for the coordinate (i) and [k.sup.2] = [[B.sup.2] [a.sup.2] - [A.sup.2] [b.sup.2]]/[[a.sup.2] ([B.sup.2] - [b.sup.2])] for the coordinate (ii).
We choose two solutions in the form of a combination of Jacobian Elliptic functions
(with an appropriate phase factor).
As seen in Table 1, the other Jacobi elliptic functions are the combinations of these three Jacobian elliptic functions
. For example, as shown before we have cs([omega], m) = cn([omega], m)/sn([omega], m).
The symbols sn u and cn u denote the Jacobian elliptic functions
sine amplitude u and cosine amplitude w.