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Journal Articles Physical Review D Year : 2023

Higher order corrections to "beyond all order'' effects in a fifth order Korteweg-de Vries equation

Abstract

A perturbative scheme is applied to calculate corrections to the leading, exponentially small (``beyond all orders'') amplitude of the ``trailing'' wave asymptotics of weakly localized solitons. The model considered is a Korteweg-de Vries equation modified by a fifth order derivative term, $\epsilon^2\partial_x^5$ with $\epsilon\ll1$, (fKdV). The leading order corrections to the tail amplitude are calculated up to ${\cal{O}}(\epsilon^5)$. An arbitrary precision numerical code is implemented to solve the fKdV equation and to check the perturbative results. Excellent agreement is found between the numerical and analytical results. Our work also clarifies the origin of a long standing disagreement between the ${\cal{O}}(\epsilon^2)$ perturbative result of Grimshaw and Joshi (SIAM J. Appl. Math. 55, 124, 1995) and the numerical results of Boyd (Comp. Phys. 9, 324, 1995).

Dates and versions

hal-04038996 , version 1 (21-03-2023)

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Gyula Fodor, Péter Forgács, Muneeb Mushtaq. Higher order corrections to "beyond all order'' effects in a fifth order Korteweg-de Vries equation. Physical Review D, 2023, 107 (10), pp.105002. ⟨10.1103/PhysRevD.107.105002⟩. ⟨hal-04038996⟩
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