Generalized factorization of second-order linear partial differential operators and reflectionless wave propagation in shallow water

Abstract: This talk will be devoted to an interpretation of a recent talk by S.M. Churilov and Yu.A. Stepanyants (Reflectionless propagation of surface waves in shallow water in a channel of variable width and depth against the background of an inhomogeneous flow, researchseminars.org/talk/mmandim/35/) from the point of view of the theory of generalized factorization of differential operators ([1]).

As shown in the works of S.M. Churilov, Yu.A. Stepanyants et al. ([2]), the factorization of a second-order operator with two independent variables, which describes the propagation of waves in an inhomogeneous one-dimensional medium, into a product of first-order operators results in the appearance of a large family of solutions that describe, from a physical point of view, waves that can be considered propagating without reflection from inhomogeneities.

We will expose briefly the theory of generalized factorization of second-order partial differential operators, originating from the works of outstanding mathematicians of the 19th - early 20th century P.-S. Laplace, G. Darboux, E. Goursat and others and further developed at the end of the 20th century.

The generalized factorization theory allows us to substantially expand the class of reflectionless solutions.

References:

[1] E.I. Ganzha, S.P. Tsarev, "Classical methods of integration of hyperbolic systems and equations of the second order", 2007, KSPU (in Russian), dx.doi.org/10.13140/2.1.4535.8084 The full text is available at the link: www.researchgate.net/profile/Sergey-Tsarev/publication/235993531_Klassiceskie_metody_integrirovania_giperboliceskih_sistem_i_uravnenij_vtorogo_poradka/links/0c96051550c72803c2000000/Klassiceskie-metody-integrirovania-giperboliceskih-sistem-i-uravnenij-vtorogo-poradka.pdf

[2] Churilov, Semyon M., and Yury A. Stepanyants. "Reflectionless wave propagation on shallow water with variable bathymetry and current." Journal of Fluid Mechanics 931 (2022).

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods