Benjamin–Ono soliton-ensemble connection with Whitham modulation theory

Let $\psi_U(x,t)$, $\psi_L(x,t)$, and $\phi_L(x,t)$ be the modulation functions defined from the scaling functions of the upper- and lower-branch eigenvalues, and let $u_0^{\mathrm B},u_1^{\mathrm B},u_2^{\mathrm B}$ denote the branches of the generally multivalued solution of Burgers' equation with initial data $u_0$. Whitham modulation connection conjecture. The functions satisfy

$$\psi_U(x,t)=u_0^{\mathrm B}(x,t),$$

while in the multivalued Burgers region,

$$\psi_L(x,t)=u_2^{\mathrm B}(x,t)-u_1^{\mathrm B}(x,t),\qquad \phi_L(x,t)=\frac12\ln\left(\frac{u_2^{\mathrm B}(x,t)-u_0^{\mathrm B}(x,t)}{u_1^{\mathrm B}(x,t)-u_0^{\mathrm B}(x,t)}\right).$$

This conjecture connects the spectral modulation fields of the Benjamin–Ono soliton ensemble with Whitham modulation theory. The source reports strong numerical evidence but no direct proof, so the claim remains open.