Four step-like solutions conjecture for the KdV similarity equation

Let $u(x)$) and $y(x)$ be functions satisfying equation

, with first integrals $H_0$ and $H_1$. A solution is called step-like if it has the asymptotic behaviour described below at one of the two ends of the real line. **Four step-like solutions conjecture.** Equation

possesses four step-like solutions for any $H_0>-4$ and $H_1>0$. Two of them have the asymptotic behaviour

$$u(x)\sim \frac{H_0+4}{16x^2}+O(x^{-3}),\quad y(x)\sim -2x\pm\frac{\sqrt{-H_1}}{2x}+O(x^{-3}),\quad x\to-\infty,$$

and another two have the same asymptotic for $x\to+\infty$. The conjecture is motivated by formal asymptotic expansions and numerical experiments; rigorous proofs of the existence of regular and separatrix solutions, their asymptotics and connection formulae, and related questions remain open.