Jacobi elliptic function

Stationary (i.e. for dA/ dz = 0) localized solutions to Eq.(3.2) represent nonlinear modes in the planar waveguide and may be found in an analytical form via matching the partial solutions of Eq.(3.2) at the core/cladding boundary. The partial solutions are Jacobi elliptical function in the core and 2l rccosh — )E]/E in the cladding (the functional dependence similar to a fundamental soliton in a uniform nonlinear medium). Here is a parameter which depends on the boundary conditions. Contrary to the modes of a linear waveguide, the transverse profile of a nonlinear mode depends on the power in the mode. 

In these formulas the symbol Za(co) stands for the Bessel function, sn (to), dn ( ), cn ( ) are the Jacobi elliptic functions having the module /(xvxv) is the general solution of the ordinary differential equation... 

If is inferesfing that Refs. [2-4] did nof include fhe explicit expressions for fhe carfesian components of fhe angular momentum operator in their respective spheroconal coordinates. The Appendix of Ref. [6] does have such expressions for the spheroconal coordinates in the Jacobi elliptic function representation. The interested reader is invited to see them and understand their structure. 

Using the definitions of the Jacobi elliptic functions we have... 

A procedure similar to that used to solve equation (66) can be applied to solve equation (110). Again, the solution is given by the Jacobi elliptic functions. The third-order polynomial under the square root on the r.h.s. of (110) has the roots... 

For a finite excluded volume (v>0), the solution to the Euler-Lagrange equation must be expressed in terms of the Jacobi elliptic functions... 

These two systems are examples from non-linear physics, where the equations can be solved in terms of elliptic functions and elliptic integrals. The reader who is not familiar with these functions, which do not arise in the same way as the previously mentioned special functions, is referred to the excellent book by Whittaker and Watson [6]. In that book, the reader will see that there are two flavours of elliptic functions, the Weierstrass and Jacobi representations, three kinds of elliptic integrals, and six kinds of pseudo-periodic functions, the Weierstrass zeta and sigma functions and the four kinds of Jacobi theta functions. Of historical interest for theoretical chemists is the fact that Jacobi s imaginary transformation of the theta functions is the same as the Ewald transformation of crystal physics [7]. 

Karl Gustav Jacob Jacobi (1804-51), professor of mathematics at Konigsberg F.R.S., 1833 famous for his work on elliptic functions, dynamics, differential equations, and determinants the paper on functional determinants was published in 1841. 

The general solution for nN(t) is a periodic function oscillating between the values and max. The solution can be given in terms of the Jacobi elliptic... 

Carl Gustav Jacob Jacobi (1804-1851), German mathematical genius and the son of a banker, graduated from school at the age of 12, professor at Koenigsberg University. Jacobi made important contributions to number theory, elliptic functions, partial differential equations, and analytical mechanics. The crater Jacobi on the Moon is named after him. 

Here a denotes the maximum field amplitude, rj is the ellipticity together with the pulse-shape function g(rj), which depends on the phase rj = (ot — k r. The laser beam is characterized by the frequency co and the wave vector k with ck = co. The transversality condition implies k A — 0. For a charged point particle interacting with this external electromagnetic field, the Hamilton-Jacobi equation reads... 

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