Axisymmetric self-similar finite-time singularity solution of the Euler equations

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 Summary of some numerically obtained solutions

N	q	# nodes	ν	\(L^{(0)}\)	\(H_{1}^{(0)}\)	\(H_{2}^{(0)}\)	\(H_{3}^{(0)}\)	\(H_{4}^{(0)}\)
1	1^†	1	0.89005	5.7801	12.8062	–	–	–
2	1^§	0	1.82365	1.53912	3.62004	0.73507	–	–
2	2^§	0	1.10091	1.40061	3.13151	0.09930	–	–
2	2^†	1	1.31387	1.54258	2.84748	−1.99573	–	–
3	2^‡	2	1.92233	1.94822	7.17371	6.01831	23.8935	–
3	3^§	0	0.6306	0.94388	1.85984	−0.756781	−0.31851	–
4	2^‡	2	2.7186	2.47909	4.33713	−4.58815	4.88998	−4.4704
4	2^‡	2	3.4453	2.9636	2.2837	−16.2556	9.9944	−6.5697

^† The function L(s) has a single node. Because of the oscillations, this may be called an “excited state solution” of the dynamical system.
^§In this case, the function L(s) has no node. Because of the simplest heteroclinic trajectory, this may be called the “ground state solution” of the dynamical system.
^‡The function L(s) has two nodes corresponding to a higher “excited state”.