Exact solutions and instability analysis of a fifth-order nonlinear evolution equation exhibiting multi-scale wave structures

Abstract

Abstract A fifth-order nonlinear partial differential equation is analyzed to construct exact traveling wave solutions through the modified extended mapping method. By converting the governing equation into an equivalent ordinary differential form, the approach enables the derivation of multiple classes of analytical solutions in a unified manner. These solutions include localized structures such as bright and dark solitons, singular waveforms, periodic and singular periodic patterns, as well as rational, exponential, and elliptic function solutions represented by both Weierstrass and Jacobi formulations, in addition to combined bright–dark soliton structures. To further characterize the system, a linear perturbation framework is applied to assess stability properties. The resulting dispersion relation is examined to determine parameter regimes associated with the onset of instability and the evolution of disturbed wave states. This work provides a generalized analytical perspective for exploring complex nonlinear wave behavior in higher-order systems and offers a foundation for future studies involving multi-scale wave interactions and stability features.