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Area of Science:

  • Nonlinear Dynamics
  • Mathematical Physics
  • Wave Phenomena

Background:

  • The nonlinear Klein-Gordon equation models various physical systems.
  • Understanding wave propagation and stability is crucial in nonlinear systems.
  • Previous studies have explored solitons and breathers in related equations.

Purpose of the Study:

  • To perform modulation instability analysis on 2D and 3D nonlinear Klein-Gordon equations.
  • To construct N-breathers and high-order rogue waves.
  • To investigate the dynamic behaviors and conditions for the existence of these waves.

Main Methods:

  • Modulation instability analysis.
  • Construction of N-breathers from 2N-solitons.
  • Bilinear method combined with improved long-wave limit technique.
  • Analysis of explicit expressions for rogue waves and lumps.

Main Results:

  • Instability regions depend on dispersion and wavenumbers.
  • Breather dynamics in 2D align with modulation instability analysis.
  • General high-order rogue waves and rational solutions were obtained for 2D and 3D equations.
  • Rogue waves are shown to always exist under specific conditions.

Conclusions:

  • Modulation instability analysis provides insights into wave behavior.
  • The nonlinear Klein-Gordon equation supports complex wave structures like breathers and rogue waves.
  • Explicit solutions confirm the general existence of rogue waves in 2D and 3D systems.