On the Analytical Solutions and Conservation Laws of the Special Extended Korteweg–De Vries Equation

We study a special case of the extended Korteweg–de Vries (eKdV) equation, arising in the description of weakly nonlinear long waves with higher-order dispersive effects. The model incorporates both third- and fifth-order dispersion and quadratic nonlinearity and describes steeper and shorter waves than the classical KdV equation. First, we determine the Lie point symmetry algebra of the equation and show that it reduces to space–time translations, which in turn motivates a traveling-wave reduction. The reduced fifth-order ODE is then analyzed by means of a calibrated  (Formula presented.) -expansion ansatz. Although homogeneous balance suggests a degree  (Formula presented.)  for exact solutions, a degree- (Formula presented.)  truncation already yields three coherent families of traveling waves—hyperbolic (solitary), trigonometric (periodic), and rational—distinguished by the discriminant of the auxiliary linear equation. Using the direct multiplier method, we construct four conservation laws, corresponding to mass, momentum, energy, and a higher-order dispersion invariant, with all  (Formula presented.)  contributions retained. Direct substitution and numerical diagnostics demonstrate that, once the algebraic wave speed is imposed, the  (Formula presented.)  profiles satisfy the PDE with residuals of order  (Formula presented.)  and preserve the conserved quantities to machine precision (below  (Formula presented.)  relative variation) over extended integration domains. These results extend the known solution structure of the special eKdV equation and illustrate the effectiveness of the  (Formula presented.)  framework for higher-order dispersive models.