Figure 1

When \(a_{1} = c_{1}\sin (\varepsilon ^{3/2}t)\), \(a_{2} = c_{2}\), \(a_{3} = c_{3}\cos (\varepsilon ^{3/2}t)\), \(a_{4} = c_{4}\), the evolution of stream function field, the stream function analytic expression is Eq. (43), where \(R = k_{1}\operatorname{sech}[20\varepsilon ^{ \frac{3}{2}}(t - 9)]\), \(F = 1.5\), \(c_{0} = - 2.4\), \(c_{1} = 2.7\), \(c_{2} = 5.3\), \(c_{3} = 1.3\), \(c_{4} = 0.6\), \(\beta = 16.3\), \(b_{0} = - 18\), \(b_{1} = 5\), \(n_{0} = 5.2\), \(m = 1.4\), \(\varepsilon = 0.1\), \(k = 6\), \(k_{1} = 10\)