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Register a new userLong-time asymptotics for the focusing nonlinear Schrodinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability
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The long-time asymptotic behavior of the focusing nonlinear Schrodinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. First, the IST is formulated over a single sheet of the complex plane without introducing a uniformization variable. The solution of the focusing NLS equation with nonzero boundary conditions is thus associated with a suitable matrix Riemann-Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well-known modulational instability within the context of the IST. This growth is then removed by suitable deformations of the Riemann-Hilbert problem in the complex spectral plane. Asymptotically in time, the $xt$-plane is found to decompose into two types of regions: a left far-field region and a right far-field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. In the latter region, it is also shown that the modulus of the leading order solution, which is initially obtained in the form of a ratio of Jacobi theta functions, eventually reduces to the well-known elliptic solution of the focusing NLS equation. These results provide the first characterization of the long-time behavior of generic perturbations of a constant background in a modulationally unstable medium.
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The long-time asymptotic behavior of solutions to the focusing nonlinear Schrodinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the presence of discrete spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the $xt$-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations.
We study the stability of traveling waves of nonlinear Schrodinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type sub-critical or critical nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For traveling waves without vortices (i.e. nonvanishing) of general nonlinearity in any dimension, we find the sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable and find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of above cases, we construct unstable and stable invariant manifolds.
We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schr{o}dinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize solutions.
We consider the semi-classical limit for the Gross-Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. For a cubic-quintic nonlinearity, we show that working with analytic data may be necessary and sufficient to obtain a similar result.
In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrodinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(mathbb{R})$, begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2sigma q^2(x,t)bar q(-x,t)=0,quadsigma=pm1, q(x,0)&=q_0(x). end{align*} We show that the solution can be represented by the solution of a Riemann-Hilbert problem (RH problem), and assuming no discrete spectrum, we majorly apply $barpartial$-steepest cescent descent method on analyzing the long-time asymptotic behavior of it.
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