We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude$q_o \gt 0$ as$x\to \pm \infty$ by implementing numerically the corresponding inverse scattering transform (IST). The Riemann–Hilbert problem associated with the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift–Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). The numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the$(x,t)$-plane depending on whether$|x/(2t)| \lt q_o$ or$|x/(2t)| \gt q_o$, as$t \to \infty$.

The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number$\beta \gt 1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric ‘fully localized’ or ‘lump’ solitary waves which decay to zero in all spatial directions. The full-dispersion KP-I equation is obtained by retaining the exact dispersion relation in the modelling from the water-wave problem. In this paper, we show that the FDKP-I equation also has a family of symmetric fully localized solitary waves which are obtained by casting it as a perturbation of the KP-I equation and applying a suitable variant of the implicit-function theorem.