Soliton Gas for the Nonlinear Schrodinger equation

In this thesis, we define and study the properties of solutions of a gas of solitons for the Focusing Nonlinear Schr\"odinger equation. A gas of solitons is an initial data with an infinite number of solitons that is defined via a suitable limit. A $N$ soliton solution is charactherised, via the scattering problem, by 2N spectral points, $\{z_j,\overline{z}_j\}_{j=1}^{N}$ and the corresponding norming constants $\{c_j\}_{j=1}^{N}$, with $z_i$ and $c_j$ complex numbers. We formulate the inverse spectral problem for the focusing nonlinear Schr\"odinger equation via a suitable Riemann-Hilbert problem that is amenable to the limit $N\to\infty$. Assuming that the spectral points $\{z_j\}_{j=1}^{N}$ fill uniformly some domain $\mathscr{D}$ in the upper half space complex plane, we show that in the limit $N\to\infty$ the inverse problem of a soliton gas can be described both by a Riemann-Hilbert problem or by a $\ov \pa$-problem. For particular choices of the domains $\mathscr{D}$, we prove that, there is a shielding effect and the soliton gas behaves, for $(x,t)$ in compact sets, as a finite multi-soliton solution. In particular when $\mathscr{D}$ is a disk, the soliton gas is a one-soliton solution. Next we study the case when $\mathscr{D}$ is an ellipse with foci located on the imaginary axis. We prove that the corresponding soliton gas initial data has a step-like oscillatory behaviour decaying exponentially at $+\infty$ while for $x\to -\infty$ the oscillations are described by the Jacobi elliptic function. The long-time asymptotic behaviour of such initial data depends on the ratio of the amplitude of the foci. If such ratio is above a certain threshold then the solution has oscillations described by genus one and three theta-functions in different sectors of the $(x,t)$-plane otherwise the oscillations are described by genus one, two and three theta functions. The asymptotic solution remains exponentially decreasing in the most right sector of the $(x,t)$ plane. Finally, inspired by the soliton gas solution we develop an extension of the Its-Izergin-Korepin-Slavnov theory of {\it integrable operators} acting on a domain of the complex plane. We show that the resolvent of an integrable operator $\mathcal{K}$ acting on a domain of the complex plane is obtained from the solution of a $\ov \pa$-problem. When the problem depends on auxiliary parameters, the related Malgrange one form is closed and coincide with the logarithmic derivative of the Hilbert-Carleman determinant of the operator $\mathcal{K}$. If the $\ov \pa$-problem is related to the inverse spectral problem of the soliton gas, then the Hilbert-Carleman determinant is a $\tau$-function of the Kadomtsev-Petviashvili (KP) or Nonlinear Schr\"odinger hierarchies.