2014
DOI: 10.1098/rspa.2013.0462
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Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials

Abstract: We consider a class of fully nonlinear FermiPasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order α > 1. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyse the propagation of localized waves when α is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the… Show more

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Cited by 55 publications
(43 citation statements)
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“…This approximation result is a continuum counterpart of what occurs [69,70] in the weak nonlinearity limit for pre-compressed discrete chains (2.1). It is known that a two-scale expansion of the discrete equations of motion yields a discrete version of the KdV equation.…”
Section: 2supporting
confidence: 58%
“…This approximation result is a continuum counterpart of what occurs [69,70] in the weak nonlinearity limit for pre-compressed discrete chains (2.1). It is known that a two-scale expansion of the discrete equations of motion yields a discrete version of the KdV equation.…”
Section: 2supporting
confidence: 58%
“…We address the logarithmic Korteweg-de Vries (log-KdV) equation derived in the context of solitary waves in granular chains with Hertzian interaction forces [6,7,8]:…”
Section: Introductionmentioning
confidence: 99%
“…Section 3 is devoted to the proof of the coercivity bound (1.14). As explained above, this coercivity bound implies linear orbital stability of the Gaussian solitary wave in the constrained space X c and it is assumed to be granted in [2,8]. The proof of coercivity relies on the decomposition of y in terms of the Hermite functions.…”
Section: Introductionmentioning
confidence: 99%
“…A study was conducted on a class of fully nonlinear Ferma-Pasta-Ulam (FPU) lattices, where it was found that this class exhibits a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of pre-compression [1]. A generalized KdV equation, called logarithmic KdV (log-KdV), was derived as formal asymptotic limits of the FPU lattices defined by…”
mentioning
confidence: 99%
“…It is well known that many well known methods have been established in scientific fields and engineering to determine the solutions with distinct physical structures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Examples of the methods that have been used are the Hirota bilinear method, the simplified Hirota's method, the Bäcklund transformation method, Darboux transformation, Pfaffian technique [7][8][9][10], the inverse scattering method, the Painlevé analysis, the generalized symmetry method, the subsidiary ordinary differential equation method, the coupled amplitude-phase formulation, the sine-cosine method, the sech-tanh method [13][14][15][16][17][18][19][20], the mapping and the deformation approach and many other methods [20][21][22][23][24][25][26][27].…”
mentioning
confidence: 99%