Existence of Dynamic Phase Transitions in a One-Dimensional Lattice Model with Piecewise Quadratic Interaction Potential

Schwetlick, Hartmut; Zimmer, Johannes

doi:10.1137/070711116

Cited by 22 publications

(42 citation statements)

References 10 publications

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“…and conclude that L(W ) is well defined for all W ∈ H. Moreover, (18) implies W − T [W ] ∈ L 2 , and in view of (19) we readily verify the formula for ∂L as well as the claimed Lipschitz property.…”

Section: The Action Functionalsupporting

confidence: 58%

Heteroclinic Travelling Waves in Convex FPU-Type Chains

Herrmann¹,

Rademacher²

2010

SIAM J. Math. Anal.

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We consider infinite Fermi-Pasta-Ulam-type atomic chains with general convex potentials and study the existence of monotone fronts that are heteroclinic travelling waves connecting constant asymptotic states. Iooss showed that small amplitude fronts bifurcate from convex-concave turning points of the force. In this paper, we prove that fronts exist for any asymptotic states that satisfy certain constraints. For potentials whose derivative has exactly one turning point, these constraints mean precisely that the front corresponds to an energy conserving supersonic shock of the "p-system," which is the naive hyperbolic continuum limit of the chain. The proof is achieved via minimizing an action functional for the deviation from this discontinuous shock profile. We also discuss qualitative properties and the numerical computation of fronts.

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Section: The Action Functionalsupporting

confidence: 58%

Heteroclinic Travelling Waves in Convex FPU-Type Chains

Herrmann¹,

Rademacher²

2010

SIAM J. Math. Anal.

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“…As already mentioned, the case δ = 0 has been solved in [SZ09]. The main result can be summarised as follows.…”

Section: Overview and Main Resultsmentioning

confidence: 97%

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

Herrmann¹,

Matthies²,

Schwetlick³

et al. 2013

SIAM J. Math. Anal.

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Phase transitions waves in atomic chains with double-well potential play a fundamental role in materials science, but very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewise-quadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the corresponding three-parameter family of waves persists as long as the perturbation is small and localised with respect to the strain variable. As a standard Lyapunov-Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterise the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive in a small ball in a suitable function space. Moreover, we derive a uniqueness result for phase transition waves with certain properties and discuss the kinetic relation.

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“…(a) We compute an explicit Euler step for (11) with (12), this means we update W tangential to M K via…”

Section: Numerical Solutionsmentioning

confidence: 99%

“…Unfortunately, very little is known about their existence. The only available results concern bi-quadratic potentials, which allow for solving (2) by Fourier methods [11,12], or almost bi-quadratic potentials, for which we can employ perturbation methods [13]. It remains a challenging task to find alternative, maybe variational, existence proofs for phase transition waves that apply to more general double-well potentials.…”

Section: Introductionmentioning

confidence: 99%

Ground waves in atomic chains with bi-monomial double-well potential

Herrmann

2013

Nonlinear Analysis: Theory, Methods & Applications

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Ground waves in atomic chains are traveling waves that corresponds to minimal non-trivial critical values of the underlying action functional. In this paper we study FPU-type chains with bi-monomial double-well potential and prove the existence of both periodic and solitary ground waves. To this end we minimize the action on the Nehari manifold and show that periodic ground waves converge to solitary ones. Finally, we compute ground waves numerically by a suitable discretization of a constrained gradient flow.Comment: new version with improved introduction and minor changes in the proofs; 14 pages, 1 figur

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Existence of Dynamic Phase Transitions in a One-Dimensional Lattice Model with Piecewise Quadratic Interaction Potential

Cited by 22 publications

References 10 publications

Heteroclinic Travelling Waves in Convex FPU-Type Chains

Heteroclinic Travelling Waves in Convex FPU-Type Chains

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

Ground waves in atomic chains with bi-monomial double-well potential

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