Abstract
This paper discusses the generalized play hysteresis operator in connection with the KdV equation. Results from the nonlinear semigroup theory are applied to assure the existence and uniqueness. The KdV equation with hysteresis is reduced to a system of differential inclusions and solved.
Highlights
The word hysteresis originates in the Greek word hysterein, which is translated as to be behind or to come later
The hysteresis is coupling to PDEs with hysteresis, which arise in many fields such plasticity, dynamics with friction, ferromagnetism, ferroelectricity, superconductivity, adsorption and desorption, biology, chemistry and economics
Several models of mechanical and magnetic hysteresis may be represented via analogical models, namely the rheological models in mechanics, circuital models in electromagnetism, by arranging elementary components in series and/or in parallel [12,13,14]
Summary
The word hysteresis originates in the Greek word hysterein, which is translated as to be behind or to come later. Ewing in 1885 [1] defined hysteresis as follows: When there are two quantities M and N such that cyclic variations of N cause cyclic variations of M, if the changes of M lag behind those of N, we may say that there is hysteresis in the relation of M and N. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. Several models of mechanical and magnetic hysteresis may be represented via analogical models, namely the rheological models in mechanics, circuital models in electromagnetism, by arranging elementary components in series and/or in parallel [12,13,14]. This work is in the framework of the Visintin researches on models of hysteresis phenomena and on related PDEs [5,6,16,17,18,19]
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