Polynomial-Based Kudryashov Analytical Methods for a Generalized Sharma–Tasso–Olver Equation with Non-Integer Nonlinearity

Source abstract

We study travelling-wave solutions of the generalized Sharma-Tasso-Olver (GSTO) equation $u_t+\lambda(u^{r})_x+\mu(u^{s})_{xx}+\nu u_{xxx}=0$, in which the exponents $r,s>0$ need not be integers. The classical, modified and generalized Kudryashov methods are applied to the reduced ordinary differential equation. A balance argument shows that the convective term fixes the degree of the expansion through $N=2/(r-1)$, whereas the nonlinear diffusion term requires $N=1/(s-1)$; the two requirements are compatible only when $r=2s-1$, a condition satisfied by the classical STO equation ($r=3$, $s=2$). For the representative non-integer cases $(r,s)=(2,\tfrac32)$ and $(\tfrac32,2)$ we show that bounded solutions within the polynomial ansatz exist only in the dispersive limit $\mu=0$, and we derive them in closed form: a $\operatorname{sech}^2$ solitary wave for $r=2$ and a $\operatorname{sech}^4$ solitary wave with speed $\kappa=4\nu L^2$ for $r=\tfrac32$. The generalized Kudryashov method extends both results to hyperbolic, singular periodic and rational families classified by the discriminant $\Delta=\mathcal{B}_1^2-4\mathcal{B}_0\mathcal{B}_2$, with $\kappa=4\nu\Delta$ in the case $r=\tfrac32$. Every solution reported here was checked by direct substitution into the governing equation.