论文标题
阻尼非线性schrödinger方程的cNoidal波的稳定性
Stability of Cnoidal Waves for the Damped Nonlinear Schrödinger Equation
论文作者
论文摘要
我们考虑了立方非线性schrödinger（NLS）方程，并在一个维圆环上进行线性阻尼，我们研究了某些孤立波轮廓在耗散动力学内的稳定性。众所周知，未阻止的立方NLS方程允许雅各比椭圆形函数的周期性波的家族。我们表明，尖齿波的家族是轨道稳定的。更确切地说，通过考虑到给定的c肾波的足够小的扰动，将始终与cNoidal波的进化保持密切（直至方程式的对称性），其质量会根据耗散动力学调节质量。该结果将轨道稳定性的概念扩展到了这种非汉顿进化。由于螺旋波不是阻尼NLS的精确解，因此扰动被迫脱离孤立波轮廓的家族。为了控制误差的世俗生长，我们发现了孤立波的一阶近似，该波浪考虑了耗散项。然后，我们使用合适的，指数降低的Lyapunov功能，该功能控制近似孤子周围的扰动的$ H^1 $ norm。
We consider the cubic nonlinear Schrödinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation is well known to admit a family of periodic waves given by Jacobi elliptic functions of cnoidal type. We show that the family of cnoidal waves is orbitally stable. More precisely, by considering a sufficiently small perturbation of a given cnoidal wave at initial time, the evolution will always remain close (up to symmetries of the equation) to the cnoidal wave whose mass is modulated according to the dissipative dynamics. This result extends the concept of orbital stability to this non-Hamiltonian evolution. Since cnoidal waves are not exact solutions to the damped NLS, the perturbation is forced away from the family of solitary wave profiles. In order to control this secular growth of the error, we find a first order approximation of the solitary wave that takes into account the dissipative term. Then we use a suitable, exponentially decreasing Lyapunov functional that controls the $H^1$-norm of the perturbation around the approximated solitons.
