The semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials is studied in a full scaling neighborhood D of the point of gradient catastrophe (x_0,t_0). This neighborhood contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions near the point of gradient catastrophe: i) each spike has the height 3|q_0(x_0,t_0,epsilon)| and uniform shape of the rational breather solution to the NLS, scaled to the size O(epsilon); ii) the location of the spikes are determined by the poles of the tritronquee solution of the Painleve I (P1) equation through an explicit diffeomorphism between D and a region into the Painleve plane; iii) if (x,t) belongs to D but lies away from the spikes, the asymptotics of the NLS solution q(x,t,epsilon) is given by the plane wave approximation q_0(x,t,epsilon), with the correction term being expressed in terms of the tritronquee solution of P1. The latter result confirms the conjecture of Dubrovin, Grava and Klein about the form of the leading order correction in terms of the tritronquee solution in the non-oscillatory region around (x_0,t_0). We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest descent method for matrix Riemann-Hilbert Problems and discrete Schlesinger isomonodromic transformations.
In this paper we consider the focusing Nonlinear Schrödinger (NLS) equation iε∂ t q + ε 2 ∂ 2
x q + 2|q| 2 q = 0, 3(1-1)
where x ∈ R and t ≥ 0 are space-time variable and ε > 0. It is a basic model for self-focusing and self-modulation, for example, it governs nonlinear transmission in optical fibers; it can also be derived as a modulation equation for general nonlinear systems. It was first integrated (with ε = 1) by Zakharov and Shabat [39] who produced a Lax pair for it and used the inverse scattering procedure to describe general decaying solutions (lim |x|→∞ q(x, 0) = 0) in terms of radiation and solitons. Throughout this work, we will use the abbreviation NLS to mean “focusing Nonlinear Schrödinger equation”.
Our interest in the semiclassical (zero-dispersion) limit (ε → 0) of NLS stems largely from its modulationally unstable behavior. As shown by Forest and Lee [19], the modulation system for NLS can be expressed as a set of nonlinear PDE with complex characteristics; thus, the system is ill posed as an initial value problem with the initial data (potential) in the form of a modulated plane wave. As a result, this plane wave is expected to break immediately into some other, presumably disordered, wave form when the amplitude and the phase of the potential possess no special properties.
In the case of an analytic initial data, the NLS evolution displays some ordered structure instead of the disorder suggested by the modulational instability (see [30], [37] and [9]), as can be seen on the well-known Figure 1 (from [9]). This figure depicts numerical simulations (obtained by D. Cai) for the absolute value of the solution q(x, t, ε) of the focusing NLS (1-1) with the initial data of a modulated plane wave q(x, 0, ε) = A(x)e i ε Φ(x) , where A(x) = e -x 2 and Φ (x) = -tanh x, Φ(0) = 0. Figure 1, as well as our numerical simulations shown on Figures 2,3, clearly identify several spatiotemporal regions of distinct asymptotic regimes of the q(x, t, ε) in the semiclassical limit ε → 0. These regions (called asymptotic regions) are separated by some curves in the x, t plane that are asymptotically independent of ε. They are called breaking curves or nonlinear caustics. In the very first asymptotic region (containing the axis t = 0) the solution q(x, t, ε) can be approximated by a slowly modulated plane wave q 0 (x, t, ε) = A(x, t)e i ε Φ(x,t) . Note that this approximation fails near (the first) breaking curve. A more complicated Ansatz that can be expressed in terms of Riemann Theta-functions is required to approximate modulated nonlinear 2n-phase waves in the asymptotic regions beyond the first breaking curve, where n can be 1, 2, 3, • • •.
A significant progress in the semiclassical asymptotics of the NLS (1-1) was achieved in [25] (pure soliton case) and [35] (pure radiation and radiation with solitons), where the order O(ε) approximation Figure 1: Absolute value |q(x, t, ε)| of a solution q(x, t, ε) to the focusing NLS (1-1) versus the space x and the time t coordinates from [9]. Here the potential q(x, 0, ε) = A(x)e i ε Φ(x) with A(x) = e -x 2 , Φ (x) = -tanh x, and ε = 0.02. q 0 (x, t, ε) of q(x, t, ε) was obtained in the first two asymptotic regions. The O(ε) error estimate is valid uniformly on compact subsets within the corresponding region. In both papers, the inverse scattering problem for the NLS (1-1) was cast as a two by two matrix Riemann-Hilbert Problem (RHP), whose semiclassical asymptotics was obtained through the combination of the nonlinear steepest descent method of P. Deift and X. Zhou ([13]) and the g-function mechanism ( [12]). The approximation q 0 , obtained in [25], [35], can be described in terms of some hyperelliptic Riemann surface R = R(x, t), which depends on x, t but does not depend on ε. The Schwarz symmetry of the focusing NLS implies that the branchpoints and the branchcuts of R are Schwarz symmetrical. In this context, regions of different asymptotic behavior of q(x, t, ε) corresponds to the different genera of R(x, t), and the approximation q 0 (x, t, ε) is expressed in terms of the Riemann Theta-functions of R. In the very first (genus zero) region (that contains the line t = 0), the approximation q 0 (x, t, ε) of q(x, t, ε) is expressed through the branch-point α(x, t) of R(x, t) as (see [35]) q 0 (x, t, ε) = A(x, t)e i ε Φ(x,t) = α(x, t)e -i ε (x,t) (0,0) {2 α(ξ,t)dξ+[4( α(x,τ )) 2 -2( α(x,τ )) 2 ]dτ } + O(ε), 4(1-2)
We will often refer to this q 0 as a modulated plane wave (or as genus zero) approximation of the solution q. The next asymptotic region (behind the first breaking curve), studied in [25] and [35], corresponds to R(x, t) of genus two; the corresponding approximation q 0 in this region has the form of a modulated nonlinear 2-phase wave.
The approximation formulae in the higher genera regions (genus 4, 6, etc.) are, in a certain sense, similar to that in the genus two region (the existence of such regions, though, remains a challenging question, see
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