Investigating novel optical soliton solutions for a generalized (3+1)-dimensional q-deformed equation

Abstract In this study, we investigate the (3+1) q -deformed tanh-Gordon equation due to its importance in the context of mathematical physics. It describes solitonic solutions in quantum field theory; it can sometimes be used in condensed matter physics to describe interactions between particles in magnetic materials or superconductors; it can model light propagation in nonlinear optical fibers or photonic crystals where the refractive index has a q -deformed structure; and it also can be applied in studying shock waves, turbulence and rogue waves where the deformation introduces corrections to classical wave phenomena. Utilizing the $$(\frac{{\mathfrak {G}}^{\prime }}{\omega {\mathfrak {G}}^{\prime }+{\mathfrak {G}}+r})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:msup> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mi>ω</mml:mi> <mml:msup> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>G</mml:mi> <mml:mo>+</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -expansion technique, we derive novel analytical solutions that enhance our understanding of the underlying dynamics. Additionally, we employ a finite element method (extended cubic B-spline method) to validate our analytical findings and explore the behavior of the q -deformed equation under different parameter regimes. Our results demonstrate the versatility of the q -deformed framework in generating rich optical phenomena, paving the way for future research in both theoretical and applied physics.