Modulation instability analysis and deriving soliton solutions of new nonlocal Lakshmanan-Porserzian-Daniel equation.

This study presents a comprehensive analytical exploration of the coupled nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation, a seminal model for describing wave propagation in highly nonlocal nonlinear media. By employing the powerful extended F-expansion technique, we derive a rich spectrum of exact analytical solutions. These include bright, dark, and singular solitons, alongside singular periodic, periodic, Jacobi elliptic, exponential, hyperbolic, and Weierstrass elliptic wave solutions. The diversity of these solutions elucidates the profound and intricate interplay between strong nonlocality and nonlinearity in governing wave formation and evolution. Furthermore, we perform a detailed linear stability analysis to investigate the modulation instability (MI) gain spectrum within the system. This analysis identifies the critical parameters-most notably the degree of nonlocality and coupling strength-that dictate the stability regimes and the dynamic evolution of the solitons. Our analytical findings are vividly complemented by graphical representations that illustrate the distinctive structures of the obtained solutions and the precise conditions for the onset of MI. This research provides crucial insights into the robust propagation of localized waves in integrable nonlocal systems, with direct potential applications in pioneering fields such as nonlinear optics, Bose-Einstein condensates, and photonic lattice design, where precise control over wave dynamics is paramount.