q-粘性Burgers方程：动力学对称性、激波孤子和q-半经典展开
q-Viscous Burgers' Equation: Dynamical Symmetry, Shock Solitons and q-Semiclassical Expansion
摘要：
提出了一类新的扩散项为非对称$q$ -扩张的$q$ -扩散热方程.用相对梯度变量表示时，该系统表现为$q$ -粘性Burgers方程.导出了该方程的多项式形式的广义Kampe de Feriet多项式的精确解，以及相应的动力学对称性和Bell多项式描述。我们发现这些多项式的生成函数的应用动力学对称性和Zassenhaus公式。我们构造并分析了激波孤子及其与不同$q$的相互作用。对$q$，孤子的速度从上到下是有界的，结果除了通常的Burgers孤子融合过程外，还发现了一种新的现象，即振幅较大但速度较小的孤子分裂成两个孤子.用$\ln q$幂的Bernoulli多项式给出了这些方程的q-半经典展开式。
英文摘要：
We propose new type of $q$-diffusive heat equation with nonsymmetric $q$-extension of the diffusion term. Written in relative gradient variables this system appears as the $q$- viscous Burgers' equation. Exact solutions of this equation in polynomial form as generalized Kampe de Feriet polynomials, corresponding dynamical symmetry and description in terms of Bell polynomials are derived. We found the generating function for these polynomials by application of dynamical symmetry and the Zassenhaus formula. We have constructed and analyzed shock solitons and their interactions with different $q$. We obtain modification of the soliton relative speeds depending on value of $q$.For $q< 1$ the soliton speed becomes bounded from above and as a result in addition to usual Burgers soliton process of fusion, we found a new phenomena, when soliton with higher amplitude but smaller velocity is fissing to two solitons. q-Semiclassical expansion of these equations are found in terms of Bernoulli polynomials in power of $\ln q$.
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