Bifurcation analysis and solutions of a three-dimensional Kudryashov–Sinelshchikov equation in the bubbly liquid

Lie symmetry analysis and exact explicit solutions of three-dimensional Kudryashov–Sinelshchikov equation

“…To Eq. (2), the periodic, solitary wave solutions for χ = 0, and kink wave solutions for χ = 0 are obtained by Bifurcations and phase portraits in [11].…”

Explicit Solutions to the (3+1)-Dimensional Kudryashov-Sinelshchikov Equations in Bubbly Flow Dynamics

“…Recently, Kudryashov and Sinelshchikov [5] developed a (3+1)-dimensional nonlinear evolution equation in a model of wave propagation in bubbly fluid flow. The model equation has gained a lot of attention where Bäcklund transformation and conservation laws [6], bifurcation [7], and densityfluctuation [8] analysis were carried out the important evolution equation.…”

“…We shall use modified tanh-coth to obtain many explicit exact solutions for (3+1)-dimensional Kudryashov-Sinelshchikov equations. The travelling wave solution of a special case of (1) is given in [7]. Note that when = 0, (1) reduces to their two-dimensional counterparts…”

Bifurcation of Traveling Wave Solutions of the Dual Ito Equation

“…In this paper, we will apply the method of dynamical system [19][20][21][22][23][24] from a mathematical point of view to study the traveling wave solutions of (1).…”