This paper investigates the existence and stability of constrained solitary waves for the supercritical nonlinear Kawahara equation with third order dispersion coefficient \(b>0\) . This model is a long-wave approximation of the capillary–gravity wave in an infinitely long flat-bottomed channel. The approach used in this paper is the variation and the instability index theory. First, we construct an unbounded open set \({\mathcal {O}}\) in \(H^2\) and construct a solution to the variational problem in \({\mathcal {O}}\cap \{\Vert u\Vert _2^2=\lambda \,\,\text {for some fixed}\,\,\lambda >0\}\) , moreover, we obtain the conditional orbital stability of the solution. Finally, we obtain the spectral stability of the solution using the instability index theory. The result in this paper generalises the recent results in Posukhovskyi and Stefanov (Discrete Contin Dyn Syst 40(7):4131–4162, 2020) and Han and Gao (J Geom Anal 34(10):311, 2024) on the \(b<0\) case and the critical and subcritical cases for \(b>0\) to the supercritical case for \(b>0\) , which answers Problem 6 in section 5 of Han and Gao (2024).
