For the Benjamin-Ono equation, the leading long-time term with x = O(t^{1/2}) is an explicit universal profile obtained from linearizing the self-similar profile equation, for rational initial data with generic reflection coefficient behavior at the origin.
SUN,Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds, Comm
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The Benjamin-Ono Equation in the Long-Time Limit: Linearized Self-Similar Universality
For the Benjamin-Ono equation, the leading long-time term with x = O(t^{1/2}) is an explicit universal profile obtained from linearizing the self-similar profile equation, for rational initial data with generic reflection coefficient behavior at the origin.