Theta functions, Gaussian series, and spatially periodic solutions of the Korteweg–de Vries equation

Abstract

It has been shown by Novikov [Funct. Anal. Appl. 8, 236 (1974)], Dubrovin et al. [Russian Math. Surveys 31, 59 (1976)], Lax [Commun. Pure Appl. Math. 28, 141 (1975)], McKean and van Moerbeke [Inv. Math. 30, 217 (1975)], and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions (’’polycnoidal waves’’) which can be expressed in terms of multidimensional Riemann theta functions. Here, it is shown that via Poisson summation, the Fourier series that define the theta functions can be transformed into an infinite series of Gaussian functions. Because the lowest terms of the Gaussian series generate the usual solitary waves, it is possible to intimately explore the relationship between solitary waves and these spatially periodic ’’polycnoidal’’ waves. Also, by using the Gaussian series, one can perturbatively calculate phase velocities and wave structure for the ’’polycnoidal’’ wave even in the strongly nonlinear regime for which the soliton (or multisoliton) is the lowest order approximation. It is further shown that the Fourier series and the complementary Gaussian series both converge so rapidly in the intermediate regime of moderate nonlinearity that one may loosely state that a solitary wave is almost a linear wave, and a linear wave almost a soliton. Thus, by using both series together, one can obtain a very complete description of these stable, finite amplitude, periodic solutions. For expository simplicity, this first discussion of the Gaussian series approach to ’’polycnoidal’’ waves will concentrate on the most elementary example: the ordinary ’’cnoidal’’ wave of the Korteweg–de Vries equation. The great virtue of the Poisson method, however, is that it extends almost trivially to other equations (the Nonlinear Schrödinger equation, the Sine–Gordon equation, and a multitude of others) and also to periodic solutions of these equations that are describable in terms of higher dimensional theta functions (’’polycnoidal’’ waves). The next to last section proves a number of generalizations of the theorems of Hirota [Prog. Theor. Phys. 52, 1498 (1974)], applicable both to ’’cnoidal’’ and ’’polycnoidal’’ solutions without restriction, and explains how these extensions will work.