A modified Korteweg-de Vries equation soliton gas under the nonzero background

In this paper, we consider a soliton gas of the focusing modified Korteweg-de Vries generated from the N-soliton solutions under the nonzero background. The spectral soliton density is chosen on the pure imaginary axis, excluding the branch cut Σc=[-i, i]. In the limit N→∞, we establish the Riemann-Hilbert problem of the soliton gas. Using the Deift-Zhou nonlinear steepest-descent method, this soliton gas under the nonzero background will decay to a constant background as x→+∞, while its asymptotics as x→-∞ can be expressed with a Riemann-Theta function, attached to a Riemann surface with genus-two. We also analyze the large t asymptotics over the entire spatial domain, which is divided into three distinct asymptotic regions depending on the ratio ξ=x/t. Using the similar method, we provide the leading-order asymptotic behaviors for these three regions and exhibit the dynamics of large t asymptotics.