Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit

Breather solutions are time-periodic and space-localized solutions of nonlinear dynamical systems. We show that the concept of anticontinuous limit, which was used before for proving an existence theorem on breathers and multibreather solutions in arrays of coupled nonlinear oscillators, can be used constructively as a high-precision numerical method for finding these solutions. The method is based on the continuation of breather solutions which trivially exist at the anticontinuous limit. It is quite universal and applicable to a wide class of nonlinear models which can be of arbitrary dimension, periodic or random, with or without a driving force plus damping, etc. The main advantage of our method compared with other available methods is that we can distinguish unambiguously the different breather (or multibreather) solutions by their coding sequence. Another advantage is that we can obtain the corresponding solutions whether they are linearly stable or not. These solutions can be calculated in their full domain of existence. We illustrate the techniques with examples of breather calculations in several models. We mostly consider arrays of coupled anharmonic oscillators in one dimension, but we also test the method in two dimensions. Our method allows us to show that the breather solution can be continued while its frequency enters the phonon band (it then superposes to a band edge phonon with a finite amplitude). We also test that our method works when introducing an extra time-periodic driving force plus damping. Our method is applied for the calculation of breathers in so-called Fermi - Pasta - Ulam (FPU) chains, that is, one-dimensional chains of atoms with anharmonic nearest-neighbour coupling without on-site potential. The breather and multibreather solutions are then obtained by continuation from the anticontinuous limit of an extended model containing an extra parameter. Finally, we show that we can also calculate `rotobreathers' in arrays of coupled rotators, which correspond to solutions with one or several rotators rotating while the remaining rotators are only oscillating. The linear stability analysis of the obtained time periodic solutions (Floquet analysis) of all these models will be done in a forthcoming paper.

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

[2]  Flach.  Obtaining breathers in nonlinear Hamiltonian lattices. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  S. Aubry.  The concept of anti-integrability applied to dynamical systems and to structural and electronic models in condensed matter physics , 1994 .

[4]  M. Kruskal,et al.  Nonexistence of small-amplitude breather solutions in phi4 theory. , 1987, Physical review letters.

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

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

[7]  Michel Peyrard,et al.  Nonlinear modes in coupled rotator models , 1996 .

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

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

[10]  Localized excitations in two-dimensional Hamiltonian lattices. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  S. Aubry.  Anti-integrability in dynamic and variational problems , 1995 .

[12]  Peter S. Lomdahl,et al.  The discrete self-trapping equation , 1985 .

[13]  Burlakov,et al.  Computer simulation of intrinsic localized modes in one-dimensional and two-dimensional anharmonic lattices. , 1990, Physical review. B, Condensed matter.

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

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

[16]  Movability of localized excitations in nonlinear discrete systems: A separatrix problem. , 1994, Physical review letters.

[17]  Schroeder,et al.  Anharmonic localized surface vibrations in a scalar model. , 1995, Physical review. B, Condensed matter.

[18]  J. Raimbault,et al.  Non-2kF charge density wave induced by phonon dispersion in one-dimensional Peierls conductors , 1995 .

[19]  Satyanad Kichenassamy.  Breather Solutions of the Nonlinear Wave Equation , 2017, 1709.07787.

[20]  Dario Bambusi,et al.  Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators , 1996 .

[21]  S. Aubry.  The twist map, the extended Frenkel-Kontorova model and the devil's staircase , 1983 .

[22]  Kivshar.  Intrinsic localized modes as solitons with a compact support. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  Sergej Flach.  Tangent bifurcation of band edge plane waves, dynamical symmetry breaking and vibrational localization , 1996 .

[24]  S. Aubry,et al.  A devil's staircase in the Zeeman response of the 1D half-filled adiabatic Holstein model , 1994 .

[25]  S. Aubry,et al.  Chaotic trajectories in the standard map. The concept of anti-integrability , 1990 .

[26]  J. Denzler.  Nonpersistence of breather families for the perturbed sine Gordon equation , 1993 .