Large x $x$ Asymptotics of the Soliton Gas for the Nonlinear Schrödinger Equation

Abstract

In this paper, we construct a Riemann–Hilbert problem of the soliton gas for the nonlinear Schrödinger equation, derived by taking the limit of the $n$ soliton solutions as $n\to \infty$. The discrete spectra corresponding to the soliton solutions are located in four disjoint intervals on the imaginary axis, which are symmetric about the real axis. We analyze the large $x$ asymptotics by setting the time variable $t$ to zero. Using the Deift–Zhou nonlinear steepest-descent method, we find that the large $x$ asymptotics at $t=0$ behave differently, as $x\to \infty$, the asymptotics decays to the zero background exponentially, while as $x\to -\infty$, the leading-order term can be expressed with a Riemann-theta function of genus three. In the conclusion, we expand this case to the general $N$ intervals and conjecture on the large $x$ asymptotics.