Smooth parametric dependence of asymptotics of the semiclassical focusing NLS (Q2346740)

The paper addresses the integrable one-dimensional nonlinear Schrödinger equation with the focusing sign of the nonlinearity, of the form \[ i\epsilon u_t + \epsilon^2u_{xx} + 2|u|^2u, \] with the initial conditions of the form \[ u_0(x)=-\mathrm{sech}(x)\exp(i\mu\epsilon^{-1} S(x)), \] with the phase defined by \(dS/dx = -\mathrm{tanh}\, ( x)\). It is known that this input generates bright solitons at \(\mu < 2\), the number of which is \(O(1/\epsilon)\) for small \(\epsilon\) (which corresponds to the so-called semiclassical limit), and does not generate solitons at \(\mu > 2\). Although the value of \(\mu = 2\) seems critical, in this sense, the present paper produces a proof of the smooth dependence of local wave parameters on \(\mu\), when this parameter crosses the critical value. The proof is produced by means of analysis of the corresponding Riemann-Hilbert problem, proving the smooth dependence on \(\mu\) of the branch points of the respective Riemann surface, which determines the local asymptotic form of the wave field. These analytical results are corroborated by numerical simulations.