Strong nonlinearity, anisotropy, and solitons in a lattice with holonomic constraints

• Published in: Wave motion 2019-06, Vol.89, p.104-115
• Main Authors: Rudenko, O.V., Hedberg, C.M.
• Format: Article
• Language: English
• Subjects: Anisotropic property; Anisotropy; Atoms; Control nonlinearities; Crystal lattices; Elliptic functions; Elliptic integrals; Equations of motion; Evolution equations; Holonomic constraints; Hookes law; Longitudinal waves; Muscles; Nonlinear differential equations; Nonlinear dynamics; Nonlinear equations; Nonlinearity; Partial differential equations; Periodic structures; Propagation; Rods; Solitary waves; Soliton solutions; Solitons; Strong nonlinearity; Strongly nonlinear
• ISSN: 0165-2125; 1878-433X; 1878-433X
• DOI: 10.1016/j.wavemoti.2019.01.001

The nonlinear dynamics of a crystal lattice where the atoms are positioned along parallel rods is studied. They may move only in one direction and this constraint leads to the appearance of nonlinearity even the forces between the atoms obey the linear Hooke’s law. This nonlinearity turns out to be strong. The equations of motion of the individual lattice atoms are written, and. in the continuum limit when the lattice period is small in comparison with the wavelength, a new strongly nonlinear partial differential equation is derived. The waves traveling in the direction orthogonal to the rods are purely transverse slow waves, governed by an equation of the Heisenberg type. In the direction along the rods, a fast purely longitudinal wave can propagate. In general, when the wave travels at an arbitrary angle, it is neither purely longitudinal nor transverse and the periodic structure exhibits anisotropic properties. Their velocity depends strongly on the direction of propagation and the structure exhibits properties similar to a skeletal muscle with stretched fibers. Special attention is paid to the soliton solutions of this equation and their behavior is studied. For non-stationary quasi-longitudinal waves, a new evolution equation, rich in symmetries, is derived. One of the solutions with a fixed transverse structure is described by elliptic integrals and evolves in accordance with a cubic nonlinear equation of the Klein–Gordon type. •A crystal lattice where the atoms can move only along one axis is studied.•Systems like this can easily be made artificially as meta-materials, but they also appear naturally.•There is no linear term and the nonlinearities dominate even for the smallest amplitudes.•A new partial differential equation is derived.•Purely transverse slow waves travel orthogonally, governed by an equation of the Heisenberg type.