Asymptotic behavior in the trailing edge domain of the solution of the KdV equation with an initial condition of the “threshold type”

We derive a new integral equation that linearizes the Cauchy problem for the Korteweg–de Vries equation for the initial condition of the threshold type, where the initial function vanishes as $x\to-\infty$ and tends to some periodic function as $x\to+\infty$. We also expand the solution of the Cauchy problem into a radiation component determined by the reflection coefficient and a component determined by the nonvanishing initial condition. For the second component, we derive an approximate determinant formula that is valid for any $t\ge 0$ and $x\in(-\infty,X_N)$, where $X_N\to\infty$ with the unboundedly increasing parameter $N$ that determines the finite-dimensional approximation to the integral equation. We prove that as $t\to\infty$, the solution of the Cauchy problem in the neighborhood of the trailing edge decays into asymptotic solitons, whose phases can be explicitly evaluated in terms of the reflection coefficient and other parameters of the problem.