Stability and bifurcation in a modulated Burgers system

TL;DRAbstract

The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis tau right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A\left ( \tau \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta left-pa

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