Journal of Mathematics, Physics and Mechanics

ABSTRACT

The generalized Korteweg-de Vries (gKdV) equation $u_t$ + f(u) $u_x$ + $u_{xxx}$ = 0, in which a general nonlinearity f(u) balances the third-order dispersive term $u_{xxx}$, unifies the classical KdV equation of 1895 with a large family of nonlinear evolution equations that govern unidirectional, weakly nonlinear, weakly dispersive waves. Depending on the power p in the standard power-law nonlinearity f(u) = $u^p$, the gKdV equation exhibits radically different behaviour: for p < 4 the solitary waves are orbitally stable, at p = 4 the L2-critical case exhibits blow-up in finite time, and for p > 4 the solitary waves are unstable and solutions may blow up or disperse. This review provides a comprehensive and critical assessment of the literature on the gKdV equation. We survey its derivation from shallow-water, internal-wave, and plasma models; the theory of solitary-wave existence, stability, and asymptotic stability; the integrability theory of its special cases (KdV, mKdV, Gardner) via the inverse scattering transform, Lax pairs, and Hirota bilinear methods; the well-posedness and ill-posedness theory in Sobolev spaces; the sharp results on critical-mass blow-up dynamics; numerical methods including conservative discontinuous Galerkin and spectral schemes; and the fractional, stochastic, and coupled generalizations. Particular attention is given to recent research published in 2025 and 2026. We close with open problems and future directions.

Keywords: Linear Programming; Simplex Method; Degeneracy; Degenerate Basic Feasible Solution; Stalling; Cycling; Bland’s Rule; Anti-Cycling Rules