The Benjamin-Ono Equation in the Long-Time Limit: Linearized Self-Similar Universality
classification
🧮 math.AP
nlin.SI
keywords
equationself-similarbenjamin-onolimitmathbbobtainprofilesolution
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We obtain the leading term in the solution of the Cauchy problem for the Benjamin-Ono equation in the limit $t\to+\infty$ with $x=O(t^{1/2})$. We show that the rate of decay exceeds that of self-similar solutions and obtain an explicit universal profile for the decaying solution, relating it to the linearization of the profile equation for self-similar solutions. The proof assumes a class of rational initial data $u_0$ in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$ that exhibit generic behavior of the reflection coefficient at the origin.
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