The propagation of dispersive shock waves (DSWs) is investigated in the cylindrical Gardner (cG) equation, which is obtained by employing a similarity reduction to the two-space one-time ($2{\displaystyle +}1$) dimensional Gardner-Kadomtsev-Petviashvili (Gardner-KP) equation. We consider the steplike initial condition along a parabolic front. Then, the cG-Whitham modulation system, which is a description of DSW evolution in the cG equation, in terms of appropriate Riemann-type variables is derived. Our study is supported by numerical simulations. The comparison is given between the direct numerical solution of the cG equation and the DSW solution obtained from the numerical solution of the Whitham system. According to this comparison, a good agreement is found between the solutions.

中文翻译：

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圆柱Gardner方程中周期波的非线性调制

在圆柱加德纳（cG）方程中研究了色散冲击波（DSW）的传播，该方程是通过对两次空间的一次相似性简化得到的（$2{\displaystyle +}\mathrm{1个}$）维Gardner-Kadomtsev-Petviashvili（Gardner-KP）方程。我们考虑沿抛物线前沿的阶梯状初始条件。然后，推导了cG-Whitham调制系统，该系统以适当的黎曼型变量的形式描述了cG方程中的DSW演化。我们的研究得到数值模拟的支持。比较了cG方程的直接数值解和从Whitham系统的数值解获得的DSW解。根据此比较，在解决方案之间找到了很好的协议。