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- W2042810926 abstract "We consider the multidimensional Frenkel-Kontorova model with one degree of freedom which is a variational problem for real functions on the lattice ${bf Z}^n$ . For every vector $alphain{bf R}^n$ there is a special class of minimal solutions $u_alpha:{bf Z}^nto{bf R}$ lying in finite distance to the linear function $x^{n+1}=alpha x$ with $xin{bf Z}^n$ . Due to periodicity properties of the variational problem $alpha$ is called the rotation vector of these solutions. The average action $A(alpha)$ of a minimal solution $u_alpha$ is obtained by averaging the variational sum over ${bf Z}^n$ . One shows that this average action is the same for any minimal solution with finite distance to the linear function $x^{n+1}=alpha x$ with rotation vector $alpha$ . Our main results concern the differentiability properties of $A(alpha)$ as a function of the rotation vector: Typically, $A$ is not differentiable at $alphain{bf Q}^n$ . This will be interpreted in a dual form as phase-locking. The phase $alpha(mu)$ of $muin{bf R}^n$ is defined by the unique vector in ${bf R}^n$ for which $A(alpha)-alphacdotmu$ is minimal. If one perturbs the variational principle by changing the parameter $mu$ , the non-differentiability of $A$ at $alphain{bf Q}^n$ forces the phase to be locked onto the rational value $alpha(mu)$ ." @default.
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- W2042810926 date "1998-04-01" @default.
- W2042810926 modified "2023-10-14" @default.
- W2042810926 title "Phase-locking in the multidimensional Frenkel-Kontorova model" @default.
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- W2042810926 doi "https://doi.org/10.1007/pl00004396" @default.
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