On the steadiness of symmetric solutions to higher order perturbations of KdV

Long Pei; Fengyang Xiao; Pan Zhang

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arXiv · 2505.16772

On the steadiness of symmetric solutions to higher order perturbations of KdV

Long Pei·Fengyang Xiao·Pan Zhang

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Abstract

We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW equation, we prove that classical and weak solutions with a priori symmetry must be traveling solutions. For the more complicated perturbed R-KdV-RLW equation, we classify all symmetric traveling solutions, and prove that there exists no nontrivial symmetric traveling solution of solitary type once dissipation or shoaling perturbations exist. This gives a new perspective for evaluating the suitableness of a model for water waves. In addition, this result illustrates the sharpness of the symmetry principle in [Int. Math. Res. Not. IMRN, 2009; Ehrnstrom, Holden \& Raynaud] for solitary waves.

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Long Pei, Fengyang Xiao, Pan Zhang. 2025-05-22. On the steadiness of symmetric solutions to higher order perturbations of KdV. https://arxiv.org/abs/2505.16772

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