@article{RosenauPikovskij2021,
  author    = {Rosenau, Philip and Pikovskij, Arkadij},
  title     = {Waves in strongly nonlinear Gardner-like equations on a lattice},
  series = {Nonlinearity / the Institute of Physics and the London Mathematical Society},
  volume    = {34},
  journal   = {Nonlinearity / the Institute of Physics and the London Mathematical Society},
  number    = {8},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0951-7715},
  doi       = {10.1088/1361-6544/ac0f51},
  pages     = {5872 -- 5896},
  year      = {2021},
  abstract  = {We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.},
  language  = {en}
}
@article{RosenauPikovskij2020,
  author    = {Rosenau, Philip and Pikovskij, Arkadij},
  title     = {Solitary phase waves in a chain of autonomous oscillators},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {30},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {5},
  publisher = {American Institute of Physics, AIP},
  address   = {Melville, NY},
  issn      = {1054-1500},
  doi       = {10.1063/1.5144939},
  pages     = {8},
  year      = {2020},
  abstract  = {In the present paper, we study phase waves of self-sustained oscillators with a nearest-neighbor dispersive coupling on an infinite lattice. To analyze the underlying dynamics, we approximate the lattice with a quasi-continuum (QC). The resulting partial differential model is then further reduced to the Gardner equation, which predicts many properties of the underlying solitary structures. Using an iterative procedure on the original lattice equations, we determine the shapes of solitary waves, kinks, and the flat-like solitons that we refer to as flatons. Direct numerical experiments reveal that the interaction of solitons and flatons on the lattice is notably clean. All in all, we find that both the QC and the Gardner equation predict remarkably well the discrete patterns and their dynamics.},
  language  = {en}
}
@article{RosenauPikovskij2014,
  author    = {Rosenau, Philip and Pikovskij, Arkadij},
  title     = {Breathers in strongly anharmonic lattices},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {89},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {2},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.89.022924},
  pages     = {9},
  year      = {2014},
  abstract  = {We present and study a family of finite amplitude breathers on a genuinely anharmonic Klein-Gordon lattice embedded in a nonlinear site potential. The direct numerical simulations are supported by a quasilinear Schrodinger equation (QLS) derived by averaging out the fast oscillations assuming small, albeit finite, amplitude vibrations. The genuinely anharmonic interlattice forces induce breathers which are strongly localized with tails evanescing at a doubly exponential rate and are either close to a continuum, with discrete effects being suppressed, or close to an anticontinuum state, with discrete effects being enhanced. Whereas the D-QLS breathers appear to be always stable, in general there is a stability threshold which improves with spareness of the lattice.},
  language  = {en}
}
@article{PikovskijRosenau2006,
  author    = {Pikovskij, Arkadij and Rosenau, Philip},
  title     = {Phase compactons},
  doi       = {10.1016/j.physd.2006.04.015},
  year      = {2006},
  abstract  = {We study the phase dynamics of a chain of autonomous, self-sustained, dispersively coupled oscillators. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons - solitary waves with a compact support - and kovatons - compact formations of glued together kink-antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex Ginzburg-Landau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons. A simple perturbation method is applied to study these effects. (c) 2006 Elsevier B.V. All rights reserved},
  language  = {en}
}