The special modular transformation for polycnoidal waves of the Korteweg–de Vries equation

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Abstract
The modular transformation of the Riemann theta function is used to show that the implicit dispersion relation for the N-polycnoidal waves of the Korteweg–de Vries equation has a countable infinity of branches for N≥2. Although the transformation also implies that each branch or mode can be written in a countable infinity of ways, it is also shown that there is a unique ‘‘physical’’ representation for each mode such that the parameters of the theta function can be interpreted as wavenumbers and amplitudes in the limit of either very small or very large amplitude. Unfortunately, the small amplitude ‘‘physical’’ representation is different (by a modular transformation) from the large amplitude ‘‘physical’’ representation for a given mode, but this difference explains an apparent paradox as described in the text. The general modular transformation expresses the theta function in terms of complex wavenumbers, phase speeds, and coordinates that have no physical relevance to the Korteweg–de Vries equation, but it is shown that for N≥2, there is a subgroup, here dubbed the ‘‘special modular transformation,’’ which gives a real result. This subgroup is explicitly constructed for general N and presented as a table for N=2.