A Preliminary Study of the Burgers Equation with Symbolic ...

SYMBOLIC COMPUTATION OF THE BURGERS EQUATION 231
The integer exponent of the amplitude factor, (10), embodies complex combinations of
harmonic interactions. Signifying the exponent as E = n1 + n2 + 2( j1 + j2 − 2) , several
scenarios can be explored. If only the leading terms of an interacting harmonic pair are
considered, then j1 = j2 = 1, and the exponent reduces to E = n1 + n2. Various wavenumber
pairs, (n1, n2), potentially comprise E. Obviously, the higher the value of E is, the greater
the complexity. Except when n1 = n2, each ordered pair, (n1, n2), has a counterpart pair,
(n2, n1), with the two elements in reversed order. This is evident in Table II, which shows
that the companion pairs exist in separate terms that may or may not have common signs
or multipliers. One pair of the set potentially yields an opposite sign in backscatter, which
does not in general lead to canceling effects, as seen in the table. If n1 = n2, no backscatter
is produced, as previously explained.
If other than leading terms are considered in the preceding analysis (i.e., j1 = 1 and j2 = 1),
then the scenarios involving the exponent of the amplitude factor become more complex,
but follow a similar line of reasoning. For example, assume j1 = 1 and j2 = 1 in (10).
Since n1 + n2 = E − 2( j1 + j2 − 2), setting E = 14 and j1 + j2 = 5 yields n1 + n2 = 8.
The companion harmonic pairs (i) (2, 6) and (6, 2), (ii) (3, 5) and (5, 3), and (iii) (4, 4)
satisfy the criterion n1 + n2 = 8. But for each of these, the following combinations of term
pairs, [ j1, j2], satisfy j1 + j2 = 5: [1, 4], [4, 1], [2, 3], and [3, 2] in which element order in
each pair is important. Because there are a total of 5 harmonic pairs, and 4 term pairs for each
harmonic pair, a total of 20 interactions satisfy the criterion E = 14. All of the harmonic
pairs produce outscatter to the 7th harmonic, because n1 + n2 = 8, but the harmonic pairs
(2, 6) and (6, 2) produce backscatter to the 3rd harmonic. The pairs (3, 5) and (5, 3) produce
backscatter to the 1st harmonic, and (4, 4) produces no backscatter. It is evident there is
a large and complex combination of harmonic interactions at play in the solution to the
Burgers equation.
Table III displays combinations of nonlinear harmonic interactions, (n1, n2), that potentially
produce outscatter and backscatter in the fundamental and the 10 harmonics produced
in the symbolic solution. However, the companion harmonic pairs denoted by (n2, n1) are
omitted for presentational convenience, but self-interactions denoted by n1 = n2 are included.
Note that outscatter combinations are fewer in number compared to backscatter
combinations, and that the fundamental only experiences backscatter. However, solution
TABLE III
Combinations of Harmonic Pairs Producing Outscatter and Backscatter for Fundamental
and Various Harmonics
Affected Outscatter Backscatter
harmonic combinations combinations
Fundamental None (1,2) (2,3) (3,4) (4,5) (5,6) (6,7) (7,8) (8,9) (9,10) ...
1st harmonic (1,1) (1,3) (2,4) (3,5) (4,6) (5,7) (6,8) (7,9) (8,10) (9,11) ...
2nd harmonic (1,2) (1,4) (2,5) (3,6) (4,7) (5,8) (6,9) (7,10) (8,11) (9,12) ...
3rd harmonic (1,3) (2,2) (1,5) (2,6) (3,7) (4,8) (5,9) (6,10) (7,11) (8,12) ...
4th harmonic (1,4) (2,3) (1,6) (2,7) (3,8) (4,9) (5,10) (6,11) (7,12) (8,13) ...
5th harmonic (1,5) (2,4) (3,3) (1,7) (2,8) (3,9) (4,10) (5,11) (6,12) (7,13) ...
6th harmonic (1,6) (2,5) (3,4) (1,8) (2,9) (3,10) (4,11) (5,12) (6,13) (7,14) ...
7th harmonic (1,7) (2,6) (3,5) (4,4) (1,9) (2,10) (3,11) (4,12) (5,13) (6,14) ...
8th harmonic (1,8) (2,7) (3,6) (4,5) (1,10) (2,11) (3,12) (4,13) (5,14) (6,15) ...
9th harmonic (1,9) (2,8) (3,7) (4,6) (5,5) (1,11) (2,12) (3,13) (4,14) (5,15) ...
10th harmonic (1,10) (2,9) (3,8) (4,7) (5,6) (1,12) (2,13) (3,14) (4,15) (5,16) ...