A modified Korteweg–de Vries equation soliton gas on a nonzero background

In this paper, we consider a soliton gas of the focusing modified Korteweg–de Vries generated from the $N$-soliton solutions on a nonzero background. The spectral soliton density is chosen on the pure imaginary axis, excluding the branch cut ${\Sigma }_c=\left(-i,i\right)$. In the limit $N\to \infty$, we establish the Riemann–Hilbert Problem of the soliton gas. Using the Deift-Zhou nonlinear steepest-descent method, this soliton gas on a nonzero background will decay to a constant background as $x\to +\infty$, while its asymptotics as $x\to -\infty$ 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 $\xi =\frac{x}{t}$. Using the similar method, we provide the leading-order asymptotic behaviors for these three regions and exhibit the dynamics of large $t$ asymptotics.