阶跃初值条件解的完全分类: 流体力学中广义 Gardner 方程的分析与数值验证

张岩,郝惠琴,郭睿

The Complete Classification of Solutions to the Step Initial Condition: Analysis and Numerical Verification for the Generalized Gardner Equation in Fluid Mechanics

Yan Zhang,Huiqin Hao,Rui Guo

图42 当 $ k_1 $ 变化时, $ t = 20 $ 时刻区域 1 内正向色散激波的演化行为. 当 $ k_1>0 $ 时, 从上到下的参数如下: $ u^- =5 $, $ u^+ =4 $; $ u^- =7/2 $, $ u^+ =\frac{1}{4}( 27-\sqrt{217}) $; $ u^- =8-\sqrt{29} $, $ u^+ =8-4\sqrt{2} $; $ u^- =10-\sqrt{65} $, $ u^+ =10-2\sqrt{17} $; $ u^- =\frac{1}{4}( 57-\sqrt{2689}) $, $ u^+ =\frac{1}{4}( 57-\sqrt{2737}) $; $ u^- =25-\sqrt{590} $, $ u^+ =25-\sqrt{593} $ 并且 $ k_2 =-1 $, $ d=1 $. 当 $ k_1<0 $ 时, $ u^- =-1-\sqrt{5} $, $ u^+ =-1-2\sqrt{2} $; $ u^- =-2-2\sqrt{2} $, $ u^+ =-2-\sqrt{11} $; $ u^- =-3-\sqrt{13} $, $ u^+ =-7 $; $ u^- =-4-2\sqrt{5} $, $ u^+ =-4-\sqrt{23} $ 并且 $ k_2 =-1 $, $ d=40 $.