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 soliton solutions as . 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 asymptotics by setting the time variable to zero. Using the Deift–Zhou nonlinear steepest‐descent method, we find that the large asymptotics at behave differently, as , the asymptotics decays to the zero background exponentially, while as , 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 intervals and conjecture on the large asymptotics.