The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves

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• 作者：Oleg V. Rudenko ; Claes M. Hedberg
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：85
• 期：2
• 页码：767-776
• 全文大小：1,188 KB
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X
• 卷排序：85

文摘

A modified equation of Burgers type with a quadratically cubic (QC) nonlinear term was recently pointed out as a new exactly solvable model of mathematical physics. However, its derivation, analytical solution, computer modeling, as well as its physical applications and analysis of corresponding nonlinear wave phenomena have not been published up to now. The physical meaning and generality of this QC nonlinearity are illustrated here by several examples and experimental results. The QC equation can be linearized and it describes the experimentally observed phenomena. Some of its exact solutions are given. It is shown that in a QC medium not only shocks of compression can be stable, but shocks of rarefaction as well. The formation of stationary waves with finite width of shock front resulting from the competition between nonlinearity and dissipation is traced. Single-pulse propagation is studied by computer modeling. The nonlinear evolutions of N- and S-waves in a dissipative QC medium are described, and the transformation of a harmonic wave to a sawtooth-shaped wave with periodically recurring trapezoidal teeth is analyzed.KeywordsStrongly nonlinear systemsNonlinear partial differential equationExact analytical solutionsQuadratically cubic equation Shock frontsNonlinear acoustics and turbulence Exact linearizations