Forced variable-coefficient Kawahara equations: closed-form traveling waves via tanh–coth and elliptic ansätze

We investigated a forced variable-coefficient Kawahara equation as a non-autonomous extension of a classical fifth-order dispersive model arising in nonlinear wave propagation. Such equations are relevant in physical situations where the effective nonlinearity, dispersive balance, and external input vary with time, thereby generating wave dynamics that are not captured by the standard autonomous Kawahara equation. The main objective of this work was to derive explicit traveling-wave solutions and to clarify how temporal modulation and forcing affect the amplitude, baseline, phase, and propagation of the resulting wave structures. By introducing a traveling-wave variable with time-dependent phase together with a forcing primitive, the governing equation was reduced to ordinary differential equations amenable to exact symbolic analysis. Two complementary exact-solution procedures were then developed. First, an improved tanh–coth method based on a Riccati auxiliary equation was used to construct localized real-valued traveling-wave families. Second, an elliptic ansatz in terms of the Weierstrass ℘ -function was employed to generate periodic solutions together with their solitary-wave degenerations. The obtained formulas yield explicit exact solutions and recover classical constant-coefficient Kawahara profiles as limiting cases. In addition, representative examples were examined using analytical formulas, numerical tables, and graphical diagnostics to illustrate the qualitative distinction between oscillatory forcing, decaying forcing, and the autonomous unforced regime. The results provide analytical benchmarks for forced dispersive models with time-dependent coefficients and establish a unified exact-solution framework for a broad class of non-autonomous Kawahara-type equations.