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In this paper, we consider fissionable wave solutions and lump solutions for the (2 + 1)-dimensional Sawada–Kotera equation via the Hirota bilinear form and the long wave limit method. We obtain two types of interactional solutions concluding the elastic interaction of a lump and a solitary wave and the elastic-fissionable-coexistence interaction of a lump and an (N – 2)-fissionable wave.

In this paper, we study the buckling of an incompressible hyperelastic rectangular layer due to compression with sliding end conditions. The combined series-asymptotic expansions method is used to derive two-coupled nonlinear ordinary differential equations (ODEs) governing the leading order of the axial strain W and shear strain G. Linear analysis yields the critical stress values of buckling. For the nonlinearly coupled system, by introducing a small parameter, the approximate analytical solutions for post-buckling deformations are obtained by using the method of multiple scales. The amplitude of buckling is expressed explicitly by the aspect ratio, the incremental dimensionless engineering stress and the mode of buckling. To the authors' best knowledge, it is the first time that such an analytical formula is obtained within the framework of two-dimensional field equations for nonlinearly elastic materials (including both geometric and material nonlinearity). Numerical computations of the coupled system are also carried out. Good agreements between the numerical and analytical solutions are found when the amplitudes of buckling are moderate. Finally, some energy analysis regarding material failure is made.