Hidden solitons in the Zabusky-Kruskal experiment: Analysis using the periodic, inverse scattering transform

Hidden solitons in the Zabusky–Kruskal experiment: Analysis using the periodic, inverse scattering transform Ivan C. Christov Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208-3125, USA Abstract Recent numerical work on the Zabusky–Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne’s nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corroborated, and the exact number of solitons, their amplitudes and their reference level is computed. Other “less nonlinear” oscillation modes, which are not solitons, are also found to have nontrivial energy contributions over certain ranges of the dispersion parameter. In addition, the reference level is found to be a non-monotone function of the dispersion parameter. Finally, in the case of large dispersion, we show that the one-term nonlinear Fourier series yields a very accurate approximate solution in terms of Jacobian elliptic functions. Key words: Hidden solitons, Korteweg–de Vries equation, Inverse scattering transform, Nonlinear Fourier analysis PACS: 05.45.Yv, 02.30.Ik 2000 MSC: 35Q51, 35P25 1. Introduction Four decades after the discovery of solitons by Zabusky and Kruskal (ZK) [35] through a computational experi- ment, the study of the evolution of harmonic initial data under the Korteweg–de Vries (KdV) equation on a periodic interval is far from complete [4]. Beyond the discovery [35] of the ability of the KdV equation’s (localized) traveling- wave solutions (termed ‘solitons’) to retain their “identity” (shape, speed) after collisions, modern numerical simula- tions by Salupere et al. of this paradigm equation of solitonics reveal such exotic features as “hidden” (or “virtual”) solitons [9], emergence of soliton ensembles [32] and long-time periodic patterns of the trajectories [29, 33]. These phenomena have been shown to be generic of nonlinear waves, as they occur under other governing equations as well [13, 27, 28] This raises the simple, yet quite fundamental, question: How many solitons emerge from a harmonic input? A successful approach to answering this question is based upon discrete spectral analysis [10, 30, 31]. The essence of this method is to characterize the solitary waves based on the information inherent in the pseudospectral numerical approximation of the underlying partial differential equation (PDE) [26]. In general, solitary wave identification is a difficult problem [15, 36], especially when multiple wave interactions occur and long time scales are considered. Fortunately, in the case of the ZK experiment, the KdV equation has an advantage over other nonlinear wave equations in that it is integrable, i.e., it can be solved exactly (in theory) using the inverse scattering transform [1] on both the infinite line and on a periodic interval. In this respect, Osborne’s nonlinear Fourier analysis [16] provides a natural framework for applying the periodic, inverse scattering transform (PIST) for the KdV equation to real-world problems. In particular, it overcomes the difficulty of the PIST being only a theoretical tool by providing a practical numerical implementation of it [17, 19, 24]. Osborne and Bergamasco [21] employed this approach to successfully reproduce the numerical results of Zabusky and Email address: christov@alum.mit.edu (Ivan C. Christov) URL: http://alum.mit.edu/www/christov (Ivan C. Christov) Preprint submitted to Mathematics and Computers in Simulation October 29, 2018 arXiv:0910.3345v2 [nlin.PS] 30 May 2010 Kruskal [35]. They were able to confirm the number of solitons observed emerging from a harmonic input and the recurrence time of the initial condition. In present work, we employ the same approach to corroborate the numerical results of Salupere et al. regarding hidden solitons. Specifically, the aim here is to determine the hidden modes’ amplitudes and classify them in the hierarchy of solutions to the periodic KdV equation (i.e., as either solitons, cnoidal waves or harmonic waves). Conceptually, one may consider this approach as a generalization of some of the ideas in [10, 31]. This paper is organized as follows. In Sec. 2, the physical form of the Korteweg–de Vries equation and its transformation to the form considered in [35] is presented. In Sec. 3, the interpretation of the PIST as nonlinear Fourier analysis is discussed. In Sec. 4, the PIST spectrum of the harmonic initial condition for the KdV equation is shown for various values of the dispersion parameter, and the hidden soliton hypothesis is discussed. Sec. 5, in the large-dispersion case, illustrates in more detail how a nonlinear Fourier series is constructed using the PIST. Finally, before concluding in Sec. 7, in Sec. 6 we elaborate on the notion of a soliton reference level for the periodic problem and its dependence on the dispersion parameter. 2. Position of th

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