A novel technique for approximate N-dual-waves solutions of the generalized nonautonomous-Klein-Fock-Gordan equation - Abdel-Gawad, Hamdy I.; Yusuf, Abdullahi (De Gruyter Poland Sp Z O O, 2026)
The Klein-Fock-Gordon equation (KFGE) is a fundamental relativistic wave equation central to quantum mechanics and quantum field theory. It plays an important role in modeling particle dynamics during intense laser-plasma interactions. The nonautonomous KFGE (NAKFGE) was not considered in the literature. This may be argued to the presence of time-dependent parameters, as, in this case, the derivation of solutions is not straight forward. Here, the NAKFGE is studied for the first time. This study pursues two primary objectives, the derivation of exact self-similar solutions of the NAKFGE for a single wave structure, and the development of a new technique for multiple solutions employing the extended unified method (EUM). The exact analytical solutions obtained provide deeper insight into plasma wave dynamics and their relevance to fusion phenomena. In particular, the results reveal strong wave-fusion behavior when the nonlinearity index n lies within the interval 0 < n < 3. This fusion predominantly occurs in the positive spatial domain, indicating favorable conditions for effective energy confinement. Moreover, the estimated hydrogen-plasma temperature reaches approximately 1.5 & times; 10(8) K, exceeding the threshold required for thermonuclear fusion. Further, these results highlight the essential influence of the gain coefficient on both the plasma temperature and the resulting wave-fusion structures. The study also emphasizes the critical role of initial stability: surpassing certain diffusion-coefficient thresholds can drive the system toward instability. Indeed this work provides important theoretical framework for future experimental and numerical investigations aimed at achieving controlled thermonuclear fusion. The intricate interplay among wave characteristics, nonlinear effects, and stability emerges as a key factor in advancing our understanding of the fusion process.
Keywords: Klein-Fock-Gordan equation; thermonuclear fusion; plasma temperature; wave fusion; extended unified method; 52.30.-q P; 52.27.Ny; 52.38.-r; 52.35.Mw