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43 当 $ k_1 $ 变化时, $ t =5 $ 时刻区域 1 内反向稀疏波的演化行为. 当 $ k_1>0 $ 时, (a) 的参数从上到下依次为: $ u^- =-3 $, $ u^+ =-7 $; $ u^- =\frac{1}{2}( -7-\sqrt{37}) $, $ u^+ =\frac{1}{2}( -7-\sqrt{133} ) $; $ u^- =\frac{1}{2}( -9-\sqrt{69}) $, $ u^+ =\frac{1}{2}( -9-\sqrt{165}) $; $ u^- =-6-\sqrt{33} $, $ u^+ =-6-\sqrt{57} $; $ u^- =\frac{1}{2}( -15-\sqrt{213}) $, $ u^+ =\frac{1}{2}( -15-\sqrt{309}) $; $ u^- =\frac{1}{4}( -35-\sqrt{1177}) $, $ u^+ =\frac{1}{4}( -35-\sqrt{1561}) $ 并且 $ k_2 =2 $, $ d=-5 $. 当 $ k_1<0 $ 时, (b) 的参数从上到下依次为: $ u^- =15-\sqrt{222} $, $ u^+ =15-\sqrt{246} $; $ u^- =\frac{1}{2}( 15-\sqrt{213}) $, $ u^+ =\frac{1}{2}( 15-\sqrt{309}) $; $ u^- =5-\sqrt{22} $, $ u^+ =5-\sqrt{46} $; $ u^- =\frac{1}{2}( 7-\sqrt{37}) $, $ u^+ =\frac{1}{2}( 7-\sqrt{133}) $; $ u^- =\frac{1}{4}( 11-\sqrt{73}) $, $ u^+ =\frac{1}{4}( 11-\sqrt{457}) $; $ u^- =1 $, $ u^+ =-3 $ 并且 $ k_2 =2 $, $ d=-5 $.