Electromagnetic knots Cauchy data

Here we summarize a program to find explicitly the Cauchy data of electromagnetic knots [25,27,30-32]. Let < 50, 00 —> S2 be two applications satisfying the following two conditions ... 

Summarizing this subsection, the group-theoretic techniques allow us to obtain three maps S3 —>. S 2 whose velocity vectors are mutually orthogonal, and with the same linking number. Next, we have to build the Cauchy data of the electromagnetic knots based on these maps. 

Consequently, we have obtained the Cauchy data of an electromagnetic knot, a representative of the homotopy class C, for which, according to (63)... 

To find the electromagnetic knot, defined at every time, from the Cauchy data (91), we use the Fourier analysis. The magnetic and electric fields can be written as... 

For the electromagnetic knot with Cauchy data given by (91), we find... 

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