Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport

Isanka Garli Hevage; Akif Ibragimov

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arXiv · 2608.04177

Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport

Isanka Garli Hevage·Akif Ibragimov

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Abstract

We develop a nonlinear continuum transport model describing the formation of localized traveling structures in a coupled two-phase medium. The model is derived from an Einstein-type material-balance formulation in which displacement is generated by diffusion and by the gradient of a background-dependent transport mechanism. The resulting system couples diffusion, nonlinear gradient-driven transport, and depletion of the background phase. The proposed framework is applicable to general chemotactic transport and, in particular, to problems related to the formation of oil and gas deposits. We analyze traveling-wave solutions and establish the existence of coherent traveling bands in the transport-dominated regime. The mobile phase is shown to form a unique one-hump profile for any given reference time, while the background component undergoes a positive, bounded monotone logistic-type transition between asymptotic states. An explicit representation of the traveling profile is obtained, and uniqueness is proved up to translation. We further derive the linearized perturbation operator around the traveling band and establish finite-time perturbation bounds through a maximum-principle argument. Finally, the traveling-wave system is reduced to a nonlinear third-order ordinary differential equation for the background profile, providing an alternative characterization of the coherent structure.

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Isanka Garli Hevage, Akif Ibragimov. 2026-08-04. Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport. https://arxiv.org/abs/2608.04177

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