Generalized solitary waves in a finite‐difference Korteweg‐de Vries equation

Abstract

Generalized solitary waves with exponentially small nondecaying far field oscillations have been studied in a range of singularly perturbed differential equations, including higher order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behavior of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behavior in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infiniteorder differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite-difference discretization, in which a lattice generalized solitary wave solution is found. K E Y W O R D S asymptotic analysis, solitons and integrable systemsStud Appl Math. 2019;142:359-384.wileyonlinelibrary.com/journal/sapm

Nanoptera and Stokes curves in the 2-periodic Fermi–Pasta–Ulam–Tsingou equation

“…where y 0 (ξ) and z 0 (ξ) take odd and even values of j respectively. There is a significant difference at this stage between the present analysis of the diatomic FPUT lattice, and the analysis of the diatomic Toda lattice from [27]; in the analysis of the Toda lattice, the leading order solution is known exactly, whereas for the FPUT lattice, it is approximated up to O(ǫ 3/2 ). Consequently, we have two small parameters in the system, which introduces an extra source of error into the approximation.…”

“…where a dash denotes a derivative with respect to ξ. Before considering the behaviour of this system for small δ, we must determine whether the δ = 0 system contains exponentially small oscillations in the far field due to the discrete nature of the system itself, such as those seen for the discretized Korteweg-de Vries (KdV) equation in [27]. It is straightforward to see that when δ = 0, the governing equations become…”

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