# Traveling wave solutions for the Richards equation with hysteresis

E.E. Behi-Gornostaeva, K. Mitra, B. Schweizer (Corresponding author)

### Abstract

We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive $\tau$-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.
Original language English 797-812 IMA Journal of Applied Mathematics 84 4 https://doi.org/10.1093/imamat/hxz015 Published - Aug 2019

### Fingerprint

Richards Equation
Traveling Wave Solutions
Hysteresis
Hysteresis Operators
Overshoot
Traveling Wave
Non-equilibrium
Saturation
Bounded Domain
Monotone
Oscillation
Term
Model
Interpretation

### Cite this

Behi-Gornostaeva, E.E. ; Mitra, K. ; Schweizer, B. / Traveling wave solutions for the Richards equation with hysteresis. In: IMA Journal of Applied Mathematics. 2019 ; Vol. 84, No. 4. pp. 797-812.
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Traveling wave solutions for the Richards equation with hysteresis. / Behi-Gornostaeva, E.E.; Mitra, K.; Schweizer, B. (Corresponding author).

In: IMA Journal of Applied Mathematics, Vol. 84, No. 4, 08.2019, p. 797-812.

TY - JOUR

T1 - Traveling wave solutions for the Richards equation with hysteresis

AU - Behi-Gornostaeva, E.E.

AU - Mitra, K.

AU - Schweizer, B.

PY - 2019/8

Y1 - 2019/8

N2 - We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive $\tau$-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.

AB - We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive $\tau$-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.

U2 - 10.1093/imamat/hxz015

DO - 10.1093/imamat/hxz015

M3 - Article

VL - 84

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EP - 812

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

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