Shielding of breathers for the focusing nonlinear Schrödinger equation

TL;DR

This work develops a rigorous framework to construct a breather gas for the focusing nonlinear Schrödinger equation by taking the $N\to\infty$ limit of $N$-breather solutions with poles densely filling a domain. The authors reformulate the inverse-scattering problem from discrete residues to a jump problem on contours surrounding the accumulation domain, with norming constants scaled as $1/N$ and interpolated by smooth densities, yielding a well-defined limiting solution $\psi_\infty(x,t)$. Under symmetry and theta-condition constraints, the gas can be reduced to a finite number of breathers (an $n$-breather) when the domain is a quadrature domain, with explicit spectra given by the zeros of $(z-d_0)^n-d_1=0$ and norming constants determined by boundary data. They illustrate the $n=1$ and $n=2$ cases, showing that the shielding phenomenon for soliton gases extends to breather gases and thereby broadening the applicability of shielding in integrable systems. Overall, the paper establishes a bridge between soliton-gas shielding and breather-gas shielding, offering a pathway to analyze step-like oscillatory initial data and more complex breather configurations in FNLS.

Abstract

We study a deterministic gas of breathers for the Focusing Nonlinear Schrödinger equation. The gas of breathers is obtained from a $N$-breather solution in the limit $N\to \infty$.\\ The limit is performed at the level of scattering data by letting the $N$-breather spectrum to fill uniformly a suitable compact domain of the complex plane in the limit $N\to\infty$. The corresponding norming constants are interpolated by a smooth function and scaled as $1/N$. For particular choices of the domain and the interpolating function, the gas of breathers behaves as finite breathers solution. This extends the shielding effect discovered in "M. Bertola, T. Grava, and G. Orsatti - Physical Review Letters, 130.12 (2023): 1" for a soliton gas also to a breather gas.

Figures (8)

• Figure 1: The complex plane divided as in \ref{['piano complesso diviso']} and 4 poles satisfying \ref{['symmetrie poli']}. Fixing $\zeta_1$ in $D_1^+$ determines the other three "fellow" points of the spectrum, $\zeta_2=-1/{\bar{\zeta_1}},\zeta_3= \bar{\zeta_1},\zeta_4=-{1}/{\zeta_1}$.
• Figure 2: Two different Kuznetsov-Ma breathers with $Z=2$, $\kappa=0$, $\phi=0$ (left) and $Z=3$, $\kappa=1$, $\phi=0$ (right). In both cases, $\psi(x,t)\to 1$ as $x\to\pm \infty.$
• Figure 3: Two different Tajiri-Watanabe breathers, left $Z=2$, $\kappa=0$, $\phi=0$ and $\alpha=\pi/6$ and right $Z=3$, $\kappa=0$, $\phi=0$ and $\alpha=-\pi/8$.
• Figure 4: The three possible kinds of 2-breather solutions.Top left: a KM-type 2-breather with $\zeta_1=2\mathop{\mathrm{\dot{\imath}}}\nolimits$ and $\zeta_2=3\mathop{\mathrm{\dot{\imath}}}\nolimits$. Top right: a TW-type 2-breather with $\zeta_1=1+2\mathop{\mathrm{\dot{\imath}}}\nolimits$ and $\zeta_2=-2+\mathop{\mathrm{\dot{\imath}}}\nolimits$. Bottom: a 2-breather of mixed type with $\zeta_1=2\mathop{\mathrm{\dot{\imath}}}\nolimits$ and $\zeta_2=1+\mathop{\mathrm{\dot{\imath}}}\nolimits$. In all three plots we set $C_1=1$ and $C_2=2$.
• Figure 5: The complex plane with the four domains $\mathcal{D}_1$, $\mathcal{D}_2$, $\overline{\mathcal{D}_1}$, $\overline{\mathcal{D}_2}$ and the four contours $\gamma^+_1$, $\gamma^+_2$, $\gamma^-_1$, $\gamma^-_2$

Theorems & Definitions (4)

• Remark 1
• Theorem 1: Existence of the solution $\psi_{\infty}$
• Remark 2
• Theorem 2