Nonlinear modes in coupled rotator models

Abstract We compare the properties of generalized nonlinear Klein-Gordon equations with a sinusoidal coupling (sine-lattice equations) to those of the usual nonlinear Klein-Gordon equation with harmonic coupling for three equations leading to the same sine-Gordon equation in the continuum limit. Using Hirota's bilinear method we find a strong analogy between the properties of nonlinear oscillatory modes in the three equations. Numerical simulations confirm this analogy even for very discrete cases. However we point out a fundamental difference between the harmonic and sinusoidal coupling by exhibiting a new class of localized nonlinear excitations which are specific to the sinusoidal coupling, the rotating modes. They can be created thermally. An approximate analytical solution is obtained for these modes which are intrinsically discrete and appear to be extremely stable, even in collisions with breathers. Possible implications for physics are discussed.

[1]  O. Bang,et al.  High order breather solutions to a discrete nonlinear Klein-Gordon model , 1995 .

[2]  Hoover,et al.  Canonical dynamics: Equilibrium phase-space distributions. , 1985, Physical review. A, General physics.

[3]  R. Vacher,et al.  Rayleigh Scattering in a Plastic Crystal Due to Orientational Relaxation , 1971 .

[4]  D. Thouless,et al.  Ordering, metastability and phase transitions in two-dimensional systems , 1973 .

[5]  S. Takeno,et al.  Self-Localized Anharmonic Rotational Modes of Bases in DNA , 1990 .

[6]  Page,et al.  Asymptotic solutions for localized vibrational modes in strongly anharmonic periodic systems. , 1990, Physical review. B, Condensed matter.

[7]  Dugald B. Duncan,et al.  Solitons on lattices , 1993 .

[8]  S. Takeno,et al.  A Coupled Base-Rotator Model for Structure and Dynamics of DNA Local Fluctuations in Helical Twist Angles and Topological Solitons , 1984 .

[9]  Existence of localized excitations in nonlinear Hamiltonian lattices. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  S. Takeno.  Duality of the Existence of Localized Modes (Low- and High-Frequency Modes) in Pure Nonlinear Lattices –Sine Lattices, Discrete Sine-Gordon Lattices, the Ablowitz-Ladik Lattices and One-Dimensional Lattices with Quartic Anharmonicity– , 1995 .

[11]  Conditions on the existence of localized excitations in nonlinear discrete systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  S. Takeno,et al.  Sine-Lattice Equation. II. Nearly Integrable Soliton Properties of π-Kinks and Sonic π-Kinks , 1986 .

[13]  Bishop,et al.  Entropy-driven DNA denaturation. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  S. Takeno,et al.  A Sine-Lattice (Sine-Form Discrete Sine-Gordon) Equation —One-and Two-Kink Solutions and Physical Models— , 1986 .

[15]  Willis,et al.  Exact determination of the Peierls-Nabarro frequency. , 1989, Physical Review B (Condensed Matter).

[16]  S. Nosé.  A unified formulation of the constant temperature molecular dynamics methods , 1984 .

[17]  Schmidt,et al.  Stability and motion of intrinsic localized modes in nonlinear periodic lattices. , 1992, Physical review. B, Condensed matter.

[18]  S. Takeno,et al.  Topological Solitons and Modulated Structure of Bases in DNA Double Helices A Dynamic Plane Base-Rotator Model , 1983 .

[19]  Ryogo Hirota,et al.  Exact Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons , 1972 .

[20]  Peyrard,et al.  Energy localization in nonlinear lattices. , 1993, Physical review letters.

[21]  S. Takeno.  Theory of Stationary Anharmonic Localized Modes in Solids , 1992 .

[22]  P. Byrd,et al.  Handbook of Elliptic Integrals for Engineers and Physicists , 2014 .

[23]  L. Petrakis,et al.  Rotational Transition and Self‐Diffusion in Polycrystalline Succinonitrile , 1963 .

[24]  Robert S. MacKay,et al.  Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators , 1994 .

[25]  S. Takeno,et al.  Sine-Lattice Equation. III. Nearly Integrable Kinks with Arbitrary Kink Amplitude , 1987 .

[26]  A. Sievers,et al.  Intrinsic Localized Vibrational Modes in Anharmonic Crystals : Stationary Modes , 1988 .

[27]  Bishop,et al.  Dynamics and thermodynamics of a nonlinear model for DNA denaturation. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  Michel Peyrard,et al.  Kink dynamics in the highly discrete sine-Gordon system , 1984 .

[29]  A. Sievers,et al.  Intrinsic localized modes in anharmonic crystals. , 1988, Physical review letters.

[30]  C. R. Willis,et al.  Localized breather-like solution in a discrete Klein-Gordon model and application to DNA , 1982 .

[31]  T. Munakata,et al.  Kink Dynamics in the Discrete Sine-Gordon System A Perturbational Approach , 1982 .