Two kinds of important bifurcation phenomena of nonlinear waves in a generalized Novikov–Veselov equation

Existence and bifurcation of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion

“…In the present paper, to better understand the role of nonlinear dispersion in the formation of patterns in liquid drops, we study dynamics of traveling wave solutions to (1.1) with m = 2, n > 1. By using the bifurcation method of dynamical systems [20][21][22][23][24][25][26][27][28][29], we shall establish the existence and bifurcation of compacton, peakon, smooth solitary wave, singular cusp, and smooth periodic wave solutions, which extend some results obtained in [30] where m = 1.…”

“…By using the qualitative theory of differential equations [22,27,29], we can see that if J(𝜙 i , 0) < 0, the equilibrium point (𝜙 i , 0) is a saddle; if J(𝜙 i , 0) > 0, the equilibrium point (𝜙 i , 0) is a center; if J(𝜙 i , 0) = 0 and the index of equilibrium point (𝜙 i , 0) is 0, then it is a cusp. Using the above qualitative analysis, phase portraits of the system can be obtained under different parameters, as shown in Figures 1,4, and 6.…”

“…By using the qualitative theory of differential equations [22, 27, 29], we can see that if $$$ J\left({\phi}_i,0\right)&amp;lt;0 $$$, the equilibrium point $$$ \left({\phi}_i,0\right) $$$ is a saddle; if $$$ J\left({\phi}_i,0\right)&amp;gt;0 $$$, the equilibrium point $$$ \left({\phi}_i,0\right) $$$ is a center; if $$$ J\left({\phi}_i,0\right)&amp;amp;#x0003D;0 $$$ and the index of equilibrium point $$$ \left({\phi}_i,0\right) $$$ is 0, then it is a cusp. Using the above qualitative analysis, phase portraits of the system can be obtained under different parameters, as shown in Figures 1,4, and 6.…”