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

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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 $k_2 =2$ , $d=-5$ .