Modulation theory solution for nonlinearly resonant, fifth‐order Korteweg–de Vries, nonclassical, traveling dispersive shock waves

Abstract

AbstractA new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.