C. R. Willis

Localized excitations in two-dimensional Hamiltonian lattices.

We analyze the origin and features of localized excitations in a discrete two-dimensional Hamiltonian lattice. The lattice obeys discrete translational symmetry, and the localized excitations exist because of the presence of nonlinearities. We connect the presence of these excitations with the existence of local integrability of the original N degree of freedom system. On the basis of this explanation we make several predictions about the existence and stability of these excitations. This work is an extension of previously published results on vibrational localization in one-dimensional nonlinear Hamiltonian lattices (Phys.Rev.E.49(1994)836). Thus we confirm earlier suggestions about the generic property of Hamiltonian lattices to exhibit localized excitations independent on the dimensionality of the lattice.

Nonlinear response of the sine-Gordon breather to an a.c. driver

We investigate the nonlinear response of the continuum sine-Gordon (SG) breather to an a.c. driver. We use an ansatz by Matsuda which is an exact collective variable (CV) solution for the unperturbed SG breather and uses only a single CV, r(t), which is the separation between the center of masses of the kink and antikink that make up the breather. We show that in the presence of a driver with an amplitude below the breakup threshold of the breather into kink and antikink, the a.c.-driven SG is quite accurately described by the r(t), which is a solution of an ordinary differential equation for a one-dimensional point particle in a potential V(r) driven by an a.c. driver and with an r-dependent mass, M(r). That is below the threshold for breakup, the solution for a driven r(t) and the use of the Matsuda identity gives a solution for the a.c.-driven SG, which is very close to the exact simulation of the a.c.-driven SG. We use a wavelet transform to analyze the frequency dependence of the time-dependent nonlinear response of the SG breather to the a.c. driver. We find the wavelet transforms of the CV solution and of the simulation of the a.c.-driven SG are qualitatively very similar to each other and often agree quite well quantitatively. In cases of breakup of the breather into K and A, where there is no appreciable radiation of phonons, we find the CV solution is very close to the exact simulation result.

Soliton ratchets induced by excitation of internal modes.

Recently Phys. Rev. Lett. 88, 184101 (2002)]] used a symmetry analysis to predict the appearance of directed energy current in homogeneously spatially extended systems coupled to a heat bath in the presence of an external ac field E (t). Their symmetry analysis allowed them to make the right choice of E (t) so as to obtain symmetry breaking which causes directed energy transport for systems with a nonzero topological charge. Their numerical simulations verified the existence of the directed energy current. They argued that the origin of their strong rectification in the underdamped limit is due to the excitation of internal modes and their interaction with the translational kink motion. The internal mode mechanism as a cause of current rectification was also proposed by Salerno and Zolotaryuk [Phys. Rev. E. 65, 056603 (2002)]]. We use a rigorous collective variable for nonlinear Klein-Gordon equations to prove that the rectification of the current is due to the excitation of an internal mode Gamma (t), which describes the oscillation of the slope of the kink, and due to a dressing of the bare kink by the ac driver. The internal mode Gamma (t) is excited by its interaction with the center of mass of the kink, X (t), which is accelerated by E (t). The external field E (t) also causes the kink to be dressed. We derive the expressions for the dressing and numerically solve the equations of motion for Gamma (t), X (t), and the momentum P (t), which enable us to obtain the explicit expressions for the directed energy current and the ac driven kink profile. We then show that the directed energy current vanishes unless the slope Gamma (t) is a dynamical variable and the kink is dressed by the ac driver.

Kink's dynamics for a deformable substrate potential

Abstract We study the static and dynamic properties of a kink in a chain of harmonically coupled atoms subjected to a deformable double-well substrate potential. We treat intrinsically the lattice discreteness without approximation and show that in some deformation-parameter ranges each period of the PN (Peierls-Nabarro) potential consists of two wells whose minima are located respectively on a lattice site and midway between two adjacent sites of the chain. In some other parameter ranges each period of the PN potential posseses a single well whose minimum is located either on a lattice site or midway between two adjacent lattice sites. We examine the kinks dynamics by using a multiple-collective-variable treatment, that is, we derive the exact equations of motion for the collective variables X and Y — which describe respectively the center-of-mass mode and the internal mode of the kink. We numerically solve the collective variable equations of motion for the trapped and untrapped regimes of the discrete-kink motion, and show that the presence of a nonlinear internal mode makes a contribution of particular importance in the discrete-kinks dynamics. Indeed, we show that during its untrapped regime, the discrete kink can undergo one or more temporary trappings and even a reflection back over several PN wells, and relate such behaviours to the effects of the excitations of the internal mode of the kink.

Collective Coordinates and Linear Modes of the Double-Sine-Gordon Kink

We present a complete Hamiltonian treatment of a kink with an internal degree of freedom, namely the double-sine-Gordon (DSG) kink. In this formalism we assign two canonical coordinates and their associated momenta to describe the motion of the center of mass of the DSG kink and the relative motion of its two sub-kinks. We show that the canonical coordinate representing the separation of the two sub-kinks describes a nonlinear oscillatory degree of freedom. We have also used the method of supersymmetry to obtain, for the first time,the complete set of eigenfunctions of small oscillations about the 4π DSG kink. This analysis was motivated by the need for an accurate expression of the shape mode in the Hamiltonian formalism.