Solutions Of NLS Research Articles

Dynamics of subcritical threshold solutions for energy-critical NLS

In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d$ ($d\geq 5$) with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in \cite{DM} for $d=3,4,5$ and later by D. Li and X. Zhang in \cite{LZ} for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the radial assumption for $d\geq 5$. A key step is to show exponential convergence to the ground state $W(x)$ up to symmetries if the scattering phenomenon does not occur. Remarkably, an interaction Morawetz-type estimate are applied.

On the stability of the compacton waves for the degenerate KdV and NLS models

In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p p . Namely, we show that all waves are spectrally stable for 2 > p ≤ 8 2>p\leq 8 , while a single mode instability occurs for p > 8 p>8 . This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range p > 8 p>8 .

Global solution in a weak energy class for Klein-Gordon-Schrödinger system

<abstract><p>Based on the possible singularity of stationary state, we revisit the initial boundary value problem of the classical Klein-Gordon-Schrödinger (KGS) system in one space dimension. The wellposedness is established in a class of Sobolev NLS solutions together with exponentially growing KG solutions.</p></abstract>

Quasiperiodic solutions of NLS with Liouvillean frequencies

Quasi-periodic solutions with Liouvillean frequency of forced nonlinear Schr\"odinger equation are constructed. This is based on an infinite dimensional KAM theory for Liouvillean frequency.

Stabilization of small solutions of discrete NLS with potential having two eigenvalues

ABSTRACT We study the long-time behavior of small (in ) solutions of discrete nonlinear Schrödinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schrödinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.

Splitting of nonlinear-Schrödinger-equation breathers by linear and nonlinear localized potentials

We consider evolution of one-dimensional nonlinear-Schr\"odinger (NLS) two-soliton complexes (breathers) with narrow repulsive or attractive potentials (barrier or well, respectively). By means of systematic simulations, we demonstrate that the breather may either split into constituent fundamental solitons (fragments) moving in opposite directions, or bounce as a whole from the barrier. A critical initial position of the breather, which separates these scenarios, is predicted by an analytical approximation. The narrow potential well tends to trap the fragment with the larger amplitude, while the other one escapes. The interaction of the breather with a nonlinear potential barrier is also considered. The ratio of amplitudes of the emerging free solitons may be different from the 3:1 value suggested by the exact NLS solution, especially in the case of the nonlinear potential barrier. Post-splitting velocities of escaping solitons may be predicted by an approximation based on the energy balance.

The L2 Weak Sequential Convergence of Radial Focusing Mass Critical NLS Solutions with Mass Above the Ground State

Abstract We study the non-scattering $L^{2}$ solution $u$ to the radial focusing mass-critical nonlinear Schrödinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly converges to the ground state $Q$ up to scaling and phase transformation. We also give some partial results on the mass concentration phenomena of the minimal mass blow-up solution.

The Landau-Lifshitz Equation, the NLS, and the Magnetic Rogue Wave as a By-Product of Two Colliding Regular “Positons”

In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear Schrödinger equation (NLS), and is aimed at resolving an old problem: how to produce multiple-rogue wave solutions of NLS using just the Darboux-type transformations. The solutions of this type—known as P-breathers—have been proven to exist by Dubard and Matveev, but their technique heavily relied on using the solutions of yet another nonlinear equation, the Kadomtsev-Petviashvili I equation (KP-I), and its relationship with NLS. We have shown that in fact one doesn’t have to use KP-I but can instead reach the same results just with NLS solutions, but only if they are dressed via the binary Darboux transformation. In particular, our approach allows us to construct all the Dubard-Matveev P-breathers. Furthermore, the new method can lead to some completely new, previously unknown solutions. One particular solution that we have constructed describes two “positon”-like waves, colliding with each other and in the process producing a new, short-lived rogue wave. We called this unusual solution (in which a rogue wave is begotten after the impact of two solitons) the “impacton”.

On the orbital instability of excited states for the NLS equation with the <i>δ</i>-interaction on a star graph

We study the nonlinear Schr\"odinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate an orbital instability of the standing waves $e^{i\omega t}\mathbf{\Phi}(x)$ of NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $\Phi(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques standard in the stability study. We also prove orbital stability of the unique standing wave solution of NLS-$\delta$ equation with repulsive nonlinearity.

Instability in nonlinear Schrödinger breathers

We consider the focusing Nonlinear Schrodinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study the initial value problem for perturbations of the background wave in Sobolev spaces. It is well-known that the associated linear dynamics for this problem describes a phenomenon known in the literature as modulational instability, also recently related to the emergence of rogue waves in ocean dynamics. In qualitative terms, small perturbations of the background state increase its size exponentially in time. In this paper we show that, even if there is no time decay for the linear dynamics due to the modulationally unstable regime, the equation is still locally well-posed in Hs, s > 1/2. We apply this result to give a rigorous proof of the unstable character of two well-known NLS solutions: the Peregrine and Kuznetsov-Mabreathers.

Existence and Asymptotic Stability of Quasi-Periodic Solutions of Discrete NLS with Potential

We prove the existence of a two-parameter family of small quasi-periodic solutions of the discrete nonlinear Schrodinger equation (DNLS). We further show that all small solutions of DNLS decouple to one of these quasi-periodic solutions and dispersive wave. As a byproduct, we show that all small nonlinear bound states including excited states are stable.

Modified energy functionals and the NLS approximation

We consider a model equation from [14] that captures important properties of the water wave equation. We give a new proof of the fact that wave packet solutions of this equation are approximated by the nonlinear Schrödinger equation. This proof both simplifies and strengthens the results of [14] so that the approximation holds for the full interval of existence of the approximate NLS solution rather than just a subinterval. Furthermore, the proof avoids the problems associated with inverting the normal form transform in [14] by working with a modified energy functional motivated by [1] and [8].

A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS

Based on our previous work for solving the nonlinear Schrödinger equation with multichannel dynamics that is given by a localized standing wave and radiation, in this work we deal with the multichannel solution which consists of a moving soliton and radiation. We apply the modulation theory to give a system of ODEs coupled to the radiation term for describing the solution, which is valid for all times. The modulation equations are solved accurately by the proposed numerical method. The soliton and radiation are captured separately in the computation, and they are solved on the translated domain that is moving with them. Thus for a fixed finite physical domain in the lab frame, the multichannel solution can pass through the boundary naturally, which cannot be done by imposing any existing boundary conditions. We comment on the differences of this method from the collective coordinates.

An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds

We prove an abstract Implicit Function Theorem with parameters for smooth operators defined on sequence scales, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.

Super-rogue waves in simulations based on weakly nonlinear and fully nonlinear hydrodynamic equations

The rogue wave solutions (rational multibreathers) of the nonlinear Schrödinger equation (NLS) are tested in numerical simulations of weakly nonlinear and fully nonlinear hydrodynamic equations. Only the lowest order solutions from 1 to 5 are considered. A higher accuracy of wave propagation in space is reached using the modified NLS equation, also known as the Dysthe equation. This numerical modeling allowed us to directly compare simulations with recent results of laboratory measurements in Chabchoub et al. [Phys. Rev. E 86, 056601 (2012)]. In order to achieve even higher physical accuracy, we employed fully nonlinear simulations of potential Euler equations. These simulations provided us with basic characteristics of long time evolution of rational solutions of the NLS equation in the case of near-breaking conditions. The analytic NLS solutions are found to describe the actual wave dynamics of steep waves reasonably well.

Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS

We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation (NLS) in C_t L^2(T), without using any auxiliary function space. This allows us to construct weak solutions of NLS in C_t L^2(T)$ with initial data in L^2(T) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in H^s(T) for s \geq 1/6.

Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator

We establish uniform error estimates of finite difference methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter $\varepsilon$ ($\varepsilon\in(0,1]$). When $\varepsilon\to0^+$, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i.e., $0<\varepsilon\ll1$, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with $O(\varepsilon^2)$-wavelength at $O(\varepsilon^4)$ and $O(\varepsilon^2)$ amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in $\varepsilon$ of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in $\varepsilon$, at the order of $O(h^2+\tau)$ and $O(h^2+\tau^{2/3})$ with time step $\tau$ and mesh size $h$ for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the $l^2$-norm and discrete semi-$H^1$ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine $\varepsilon$-dependent error estimates of NLSW, $\varepsilon$-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-$H^1$ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.

Scattering for solutions of NLS in the exterior of a 2D star–shaped obstacle

We prove that solutions to non-linear Schrodinger equations in two dimensions and in the exterior of a bounded and smooth star-shaped obstacle scatter in the energy space. The non-linear potential is defocusing and grows at least as the quintic power.

CLOSED FORM SOLUTIONS TO THE INTEGRABLE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using a matrix triplet (A, B, C). Here A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N-soliton and breather solutions as special cases. We also prove that these solutions reduce to known continuous matrix NLS solutions as the discretization step vanishes.

Blow-up profile to solutions of NLS with oscillating nonlinearities

This paper is concerned with the blow-up solutions of nonlinear Schrödinger equation (NLS) with oscillating nonlinearities. The limiting profiles of the blow-up solutions u(t, x) with initial data \({\|u_0\|_{L^2}=\|Q\|_{L^2}}\) are obtained. It reads that \({|u(t,x)|^2\rightarrow \|Q\|_{L^2}^2\delta_{x=y_1}}\) (Dirac function), as \({t \rightarrow T}\) , and that u(t, x) converges strongly to Q(x) in the energy space \({\Sigma=\{u\in H^1; \int |x|^2|u|^2dx<\infty\}}\) up to scaling and phase parameters and also translation in the nonradial case.