In this work, we investigate the nonlinear p-defocusing Klein–Gordon equation with a general defocusing nonlinearity , where \( p > 1\) , in two spatial dimensions. We use Lie symmetry analysis to systematically reduce the equation to simpler forms, which allows us to discover invariant solutions and facilitate complex stability analysis. Using Lie group methods, we derive several classes of reduced equations and exact travelling wave solutions. The solutions exhibit localised solitary waves, modulated localised structures (spatially confined waves with internal oscillations), and periodic wave trains depending on the balance between the wave speed, direction, and the nonlinearity parameter p. The stability of these travelling wave solutions is analysed using a linear stability framework that identifies conditions under which the solutions are stable or prone to growth and decay. Dispersion relations, phase velocities, and group velocities are derived to characterise the wave behaviour in detail. We also study the physical implications of the solutions in plasma physics and nonlinear wave dynamics, presenting graphical representations to illustrate their profiles and stability characteristics.
