Russian Physics Journal, Vol. 61, No. 2, June, 2018 (Russian Original No. 2, February, 2018)
ANALYSIS OF SOME PROPERTIES OF THE NONLINEAR
SCHRÖDINGER EQUATION USED FOR FILAMENTATION
A. A. Zemlyanov and A. D. Bulygin UDC 530.182.551.510.42 + 535.621.33
Properties of the integral of motion and evolution of the effective light beam radius are analyzed for the
stationary model of the nonlinear Schrödinger equation describing the filamentation. It is demonstrated that
within the limits of such model, filamentation is limited only by the dissipation mechanisms.
Keywords: self-focusing, filamentation, virial theorem, integrals of motion.
The mathematical model describing the phenomenon of self-focusing and filamentation is the nonlinear
Schrödinger equation (NSE). In the stationary variant, it has dimensionality (2 + 1). It should be noted that unlike the
NSE with dimensionality (1 + 1) for which the theory of the inverse problem of scattering was constructed in [1, 2] and
exact soliton solutions were obtained for the NSE with cubic local nonlinearity in higher-order dimensionalities, exact
solutions for the NSE studied in the present work are lacking. Nevertheless, various approximate and asymptotic
methods of solving the NSE, including equations with nonlocal nonlinearity [3, 4], are actively developed. However, for
the NSE with local cubic nonlinearity and higher-order dimensionality, only a limited set of exact analytical results has
been known. Among them are the property of the global collapse in the Kerr medium [1, 5] and the Townes solution
that is non-analytic  and unstable according to the Bespalov–Talanov instability theory .
As demonstrated below, a number of simple and useful formulas for the integral parameters can be derived for
a more general case of local nonlinearity of the form  considered, for example, in [8, 9] devoted to the filamentation.
It is surprising that in the literature on self-focusing and filamentation, these trivial formulas do not receive due
attention [5, 10]. The reason is that in connection with the development of modern highly efficient numerical methods,
the filamentation problem is mainly investigated based on numerical experiments [5, 10]. In addition, as already
indicated above, the approximate analytical methods of solving this problem, including the variation method [9, 11] and
the methods of renormalization group analysis , are actively developed.
However, these approximate methods use either the approximation of trial functions [9, 11] or the eikonal
approximation  (this approximation, as demonstrated in [13, 14], is incorrect for the filamentation process). In our
opinion, the choice of the localized functions such as Gaussian or similar ones is not physical, since it allows one
neither to describe the formation of the ring structure typical of the filamentation, nor the increase of the integral light
beam size established in . As to the direct numerical modeling, it is carried out blindly, as a rule, without testing on
model solutions that are lacking. For the stationary problem formulation (when the motion integrals are known ), as
demonstrated in , such blind approach (realized by numerical schemes standard for investigation of the
filamentation) leads to significant distortion of the results for some important physical quantities.
V. E. Zuev Institute of Atmospheric Optics of the Siberian Branch of the Russian Academy of Sciences,
Tomsk, Russia, e-mail: firstname.lastname@example.org; email@example.com. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika,
No. 2, pp. 136–141, February, 2018. Original article submitted July 13, 2017; revision submitted November 13, 2017.
1064-8887/18/6102-0357 2018 Springer Science+Business Media, LLC