2026/01/01 by Mark A. Hoefer, Anna Vainchtein
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons #cond-mat.mtrl-sci #nlin.PS
paper · pdf · doi:10.1017/jnw.2026.10042
42 pages, 14 figures
openalex publication_date 2026/01/01 · arxiv created 2026/06/16 · openalex created_date 2026/06/29 · openalex updated_date 2026/08/01 · arxiv updated 2026/08/03
Abstract A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi–Pasta–Ulam lattice with a general cubic interaction force. Explicit periodic travelling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic travelling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic velocities is shown to lead to modulational instability of periodic travelling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic travelling waves and initial value problems that also reveal additional short-wavelength instabilities.