Solitary‐Wave Interactions in Elastic Rods

Abstract

The propagation of longitudinal deformation waves in an elastic rod is modelled by the nonlinear partial differential equationwith p = 3 or 5. This equation is first derived under a range of possible constraints. We then show that this equation and even certain generaliza.ions do not pass the Painleve test, and hence are probably not completely integrable. Finally, we study the head-on collision of two equal solitary waves numerically and also asymptotically for small and large amplitude.

On small amplitude solutions to the generalized Boussinesq equations

“…The equations (1.1) and (1.2) also cover another various physical phenomena such as the dynamics of stretched string [30,9], Fermi-Pasta-Ulam problems [10], the evolution of long internal waves of moderate amplitude [1], nonlinear Alvén waves [27] and so on.…”

Asymptotic behavior for a semilinear second order evolution equation

“…For each fixed ε 0 > 0, equation (E ε 0 ) is a special form of the so-called improved Boussinesq equation (see [3,19,20,31]) with damped term −Δu t , which was used to describe ion-sound waves in plasma by Makhankov [20,21] and also known to represent other sorts of 'propagation problems' of, for example, lengthways waves in nonlinear elastic rods and ion-sonic waves of space transformations by a weak nonlinear effect (see [3,10]). …”

Local well posedness, asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time

“…The equations that fall into this class are known to represent some sort of 'propagation problems' (see [4,6,22]; also [19] and the references therein), among which a specific problem is (1.3)…”

“…Here −A : D(A) ⊂ X → X is the generator of an exponentially decaying analytic semigroup of bounded linear operators in X and X 1 is the domain of A with the graph norm. The equations that fall into this class are known to represent some sort of 'propagation problems' (see [4,6,22]; also [19] and the references therein), among which a specific problem is Key words and phrases. Evolution equations of the second order in time, existence, uniqueness and continuous dependence of solutions on initial conditions, asymptotic behavior of solutions, attractors, regularity, critical exponents.…”