扰动Korteweg-de Vries方程的解析和数值研究
Analytical and numerical studying of the perturbed Korteweg--de Vries equation
摘要：
考虑扰动的Korteweg-de Vries方程。该方程用于描述一维粘性气体动力学、含气泡液体中的非线性波和非线性声波。利用Painlevé方法研究了该方程的可积性。建立了扰动Korteweg-de Vries方程可积的参数条件。找到了该方程所允许的新的经典和非经典对称性，得到了所有相应的对称性约化。构造了这些约简的新的精确解。它们通过三角函数和艾里函数表示。数值研究了扰动Korteweg-de Vries方程精确解的稳定性。
英文摘要：
The perturbed Korteweg--de Vries equation is considered. This equation is used for the description of one--dimensional viscous gas dynamics, nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves. The integrability of this equation is investigated using the Painlevé approach. The condition for parameters for the integrability of the perturbed Korteweg--de Vries equation equation is established. New classical and nonclassical symmetries admitted by this equation are found. All corresponding symmetry reductions are obtained. New exact solutions of these reductions are constructed. They are expressed via trigonometric and Airy functions. Stability of the exact solutions of the perturbed Korteweg--de Vries equation is investigated numerically.
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