Moving Boundary Problems for Heterogeneous Media. Integrability via Conjugation of Reciprocal and Integral Transformations

Abstract

The combined action of reciprocal and integral-type transformations is here used to sequentially reduce to analytically tractable form a class of nonlinear moving boundary problems involving heterogeneity. Particular such Stefan problems arise in the description of the percolation of liquids through porous media in soil mechanics.

Modulated Kepler-Ermakov triads. Integrable Hamiltonian structure and parametrisation

“…Application has been subsequently been made to reduce Ermakov-modulated nonlinear Schrödinger models [10][11][12][13] and coupled solitonic sine-Gordon, Demoulin and Manakov systems [14] to their integrable counterparts. In [15,16] modulated systems have been recently arisen in the analysis of heterogeneous moving boundary problems such as model the transport of liquids through a porous medium such as soil.…”

Moving boundary problems for a canonical member of the WKI inverse scattering scheme: conjugation of a reciprocal and Möbius transformation

“…Nonlinear moving boundary problems involving heterogeneity arise naturally in the analysis of water transport through porous media such as in soil mechanics [25,26,70,71]. In the present solitonic context, a class of involutory transformations is applied to a canonical Stefan-type system of the kind (5) to construct an associated solvable class of inhomogeneous Dym moving boundary problems.…”

A Class of Moving Boundary Problems with a Source Term. Application of a Reciprocal Transformation

“…In [1,2,8,9,11], a novel integral representation version of the Hopf-Cole transformation was used to treat certain classes of Stefan-type problems for Burgers equation. Reciprocal-type transformations on the other hand have been previously applied to solve a wide range of nonlinear moving boundary problems such as arise in nonlinear heat conduction, the analysis of the melting of metals, sedimentation and other physical contexts [16,[20][21][22][23][24][25][26][27][28][29][30][31]. It is remarked that the results in [27][28][29] obtained via application of reciprocal transformations concern moving boundary problems for certain solitonic equations, namely, the Dym, potential mkdV as well as the extended Dym equation derived via geometric considerations in [32].…”