Investigation of Boussinesq-Type Special Solitons via Extended Hirota Bilinear Method with Spectral Tunability

Understanding nonlinear wave dynamics is important in mathematical physics because of their relevance to a broad range of physical applications. Solitons are localized wave structures that maintain their shape during propagation through a balance between nonlinear and dispersive effects. In this work, we study exact soliton solutions of a Boussinesq-type (BT) system and analyze the resulting wave structures. Although the conventional Hirota bilinear method (HBM) is effective for obtaining exact multi-soliton and breather solutions of nonlinear evolution equations (NLEEs), it has limited flexibility in describing a wider range of localized structures. To address this limitation, the Extended Hirota bilinear method (EHBM) is applied to the BT equation to obtain singular and non-singular soliton structures, including multi-singular solitons, breathers, quasi-breathers, lump waves, and hybrid states, using suitable real and complex spectral parameters. The three-dimensional surface plots and contour maps illustrate the spatial and temporal behaviour of these solutions. Different choices of the spectral parameters provide tunability of the resulting wave profiles within the EHBM framework.