Abstract
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.
Highlights
The Burgers equation was until recently the only known nonlinear partial differential equation of the second order which simultaneously has the two important properties that: (i) it can be exactly linearized by a simple transformation, and (ii) it has a significant physical meaning
A second nonlinear partial differential equation which can be linearized by a simple substitution was indicated recently by the authors [6,7,8]
In this paper the attention was focused on the possibility of an exact linearization of the quadratically cubic (QC) Eq (5), on its exact solution, as well as on the behavior of shock fronts and single N-wave and S-wave pulses
Summary
The Burgers equation was until recently the only known nonlinear partial differential equation of the second order which simultaneously has the two important properties that: (i) it can be exactly linearized by a simple transformation (using the Hopf–Cole substitution), and (ii) it has a significant physical meaning. The Burgers equation adequately describes the physical phenomena of high-intensity wave propagation in dissipative non-dispersive media with quadratic nonlinearity. A second nonlinear partial differential equation which can be linearized by a simple substitution was indicated recently by the authors [6,7,8]. This is a quadratically cubic (QC) Burgers-type equation. One can say that the usual cubic nonlinearity V 3 [9,10] is modeled here by the piecewise quadratic relation |V |V This function is continuous, as is its first derivative, while the second derivative has a singularity at V = 0.
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