Exact Analytical Solutions of the Generalized KdV Equation and the Duality between Focusing and Defocusing Regimes

The generalized quadratic Korteweg–de Vries (KdV) equation admits two nonlinear travelling wave branches, yet existing exact solution studies typically treat only one sign of the nonlinearity and do not identify the correct mapping into the Weierstrass invariants. This paper derives the complete branch resolved travelling wave structure of the normalized equation reducing it to a first order cubic equation and identifying it with the canonical Weierstrass differential equation. Four structural contributions are obtained: (i) explicit travelling wave solutions for both focusing (b= −12) and defocusing (b=12) branches; (ii) the correct invariant mappings (g2,g3)=(−C1,−C2) and (C1,C2) for the two nonlinearities; (iii) exact symbolic verification of each solution using the defining identities of the Weierstrass function; and (iv) the identification of the symmetry (b,y)→(−b,−y), which provides an exact duality between the two branches without requiring a change in travelling speed. The zero invariant limit yields the rational profiles ±1/(x−t)2. The resulting branch resolved and symmetry consistent solution structure offers exact analytical benchmarks for symbolic and numerical studies of nonlinear dispersive waves.

symbolic computation; nonlinear wave equation; KdV equation; exact solutions; Weierstrass elliptic function; travelling waves; focusing and defocusing regimes.

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