Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport
Abstract
We develop a nonlinear continuum transport model for localized traveling structures in a coupled two-phase medium. The model is derived from an Einstein-type material-balance formulation combining diffusion with directed transport induced by the relative gradient of an evolving background phase. The resulting system couples diffusion, nonlinear gradient-driven transport, and irreversible background depletion. The framework provides a mathematical model for chemotactic transport and a possible modeling perspective for processes associated with the formation of oil and gas deposits. We analyze traveling-wave solutions and identify beta = chi/D = 1 as the critical threshold for globally positive localized traveling bands under prescribed asymptotic conditions as the traveling coordinate s tends to plus or minus infinity. Such bands exist for beta greater than or equal to 1, whereas for 0 < beta < 1 the positive branch reaches zero background concentration at a finite traveling coordinate. For beta greater than or equal to 1, explicit profiles are obtained and, for fixed wave speed c > 0 and right-end state V_plus > 0, are unique up to spatial translation in the traveling coordinate s. We further derive the variable-coefficient linearized perturbation operator and establish a conditional finite-time a priori estimate for the mobile-phase perturbation under bounded background forcing using a maximum-principle argument.
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BibTeX
@misc{hevage2026travelingwave,
title = {{Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport}},
author = {Isanka Garli Hevage and Akif Ibragimov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.04177},
eprint = {2608.04177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04177},
}
RIS
TY - UNPB
TI - Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport
AU - Isanka Garli Hevage
AU - Akif Ibragimov
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2608.04177
UR - https://arxiv.org/abs/2608.04177
AB - We develop a nonlinear continuum transport model for localized traveling structures in a coupled two-phase medium. The model is derived from an Einstein-type material-balance formulation combining diffusion with directed transport induced by the relative gradient of an evolving background phase. The resulting system couples diffusion, nonlinear gradient-driven transport, and irreversible background depletion. The framework provides a mathematical model for chemotactic transport and a possible modeling perspective for processes associated with the formation of oil and gas deposits. We analyze traveling-wave solutions and identify beta = chi/D = 1 as the critical threshold for globally positive localized traveling bands under prescribed asymptotic conditions as the traveling coordinate s tends to plus or minus infinity. Such bands exist for beta greater than or equal to 1, whereas for 0 < beta < 1 the positive branch reaches zero background concentration at a finite traveling coordinate. For beta greater than or equal to 1, explicit profiles are obtained and, for fixed wave speed c > 0 and right-end state V_plus > 0, are unique up to spatial translation in the traveling coordinate s. We further derive the variable-coefficient linearized perturbation operator and establish a conditional finite-time a priori estimate for the mobile-phase perturbation under bounded background forcing using a maximum-principle argument.
ER -