Solitons Perturbed By Thermal Noise In One-dimensional Nonlinear Lattice Models

It is well known that the perturbation is unavoidable in real physical world, such as impurity, constant or periodic external forces, damping,and thermal noise, so when nonlinear lattice model is applied to real physical situation, equations of motion of the system must incorporate additional perturbations among which thermal noise is typical. The soliton in nonlinear lattice model can travel without perturbations all the time, keeping invariant velocity and shape, but in the presence of thermal noise the shape of soliton changes continuously with time, and the distortion of shape has effect on the velocity which affects the shape in turn, so the soliton can not propagate freely because it will be flooded by thermal noise after a long time. It shows that the effects of thermal noise on the nonlinear excitations in lattice chains are not negligible. In this paper we perform numerical simulation for solitons on one-dimensional anharmonic monoatomic chains in the presence of thermal noise. Main work and results are as follows.We consider an anharmonic chain of particles with nearest-neighbour interactions. The particles interact via an harmonic potential with a quartic anharmonicity. After finding the equations of motion we add Stokes damping and Gaussian white noise, which have to fulfill fluctuation-dissipation theorem, to the equations of motion. Then we get the perturbed equations of motion of the system, which is simulated numerically with Heun method later, beginning with the discrete envelope soliton at rest. Firstly, the mass of the perturbed envelope soliton is calculated. Result shows that the mass of the envelope soliton decreases with time and at the same time the mass of the envelope soliton at higher temperature is bigger. Then we study the center of mass of the perturbed envelope soliton, and find that the damping dissipates the energy of system but doesn't keep the soliton from propagating and that the soliton propagates more difficultly at higher temperature. Lastly, we show how the half-time change with temprature. Here we difine the time, when the amplitude of the envelope soliton decreases by half due to damping, as"half-time". Thermal noise, inputting disordered energy into the system, prevents degeneration of the envelope soliton induced by damping, and increases the half-time of the envelope soliton relatively.Next we take into account an anharmonic chain of particles with both cubic and quartic anharmonicity besides an harmonic potential. It is found that there exists envelope solitons, kink-envelope, antikink-envelope solitons and so on. Tho se kink-envelope solitons are our subject. We use the same method above to get the perturbed equations of the system. For the sake of simplicity, we calculate the center of mass and variance of position of kink-envelope soliton which locates at Brillouin border. The center of mass of kink-envelope soliton is independent on time. That is to say, the soliton is at rest all the time. But the center of mass of kink-envelope soliton fluctuates randomly from its site of center. The variance of position of kink-envelope soliton is not linear with time any more, containing the high order of time.