arXiv subjects

Stefan Le Coz

Publications and source records attributed to Stefan Le Coz.

At least 19 recordsLinked to original sources

arXiv2026

Dispersion for the Schr{ö}dinger equation on the line with short-range array of delta potentials

We study dispersive properties of the one-dimensional Schr{ö}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ${\ell}$^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) $\rightarrow$ L^$\infty$ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{ö}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator.

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arXiv2025

Ground States for the Nonlinear Schr{ö}dinger Equation on Open Books and Dimensional Reduction to Metric Graphs

In this work, we study the dimensional reduction of stationary states in the shrinking limit for a broad class of two-dimensional domains, called open books, to their counterparts on metric graphs. An open book is a two-dimensional structure formed by rectangular domains sharing common boundaries. We first develop a functional-analytic framework suited to variational problems on open books and establish the existence of solutions as constrained action minimizers. For graph-based open books (i.e., those isomorphic to the product of a graph with an interval) we prove the existence of a sharp transition in the dimensionality of ground states. Specifically, there exists a critical transverse width: below this threshold, all ground states coincide with the ground states on the underlying graph trivially extended in the transverse direction; above it, ground states become genuinely two-dimensional.

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arXiv2025

Analysis of quasi-periodic waves of cubic nonlinear Schr{ö}dinger equations

We study the quasi-periodic standing wave solutions of the focusing and defocusing cubic nonlinear Schr{ö}dinger equations in dimension one. In the defocusing case, we establish a diffeomorphic correspondence between the invariants of the ordinary differential equation of the wave profiles and the conserved quantities of the evolution equation. We introduce a numerical scheme to compute the minimizers of the energy at fixed mass and momentum for both focusing and defocusing cases. The scheme is based on a gradient flow approach with discrete renormalization at each time step. The novelty of our scheme is that the renormalization step deals at the same time with the mass and the momentum constraints. In numerical experiments, we observe that a given solution of the profile ordinary differential equation is also a minimizer of the energy at corresponding mass and momentum.

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arXiv2025

Ground States of the Nonlinear Schr{ö}dinger Equation on the Tadpole Graph with a Repulsive Delta Vertex Condition

We consider the stationary nonlinear Schr{ö}dinger equation set on a tadpole graph with a repulsive delta vertex condition between the loop and the tail of the tadpole. We establish the existence of an action ground state when the size of the loop is either very small or very large. Our analysis relies on variational arguments, such as profile decomposition. When it exists, we study the shape of the ground state using ordinary differential equations arguments, such as the study of period functions. The theoretical results are completed with a numerical study.

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arXiv2025

Traveling waves in periodic metric graphs via spatial dynamics

The purpose of this work is to introduce a concept of traveling waves in the setting of periodic metric graphs. It is known that the nonlinear Schr{ö}dinger (NLS) equation on periodic metric graphs can be reduced asymptotically on long but finite time intervals to the homogeneous NLS equation, which admits traveling solitary wave solutions. In order to address persistence of such traveling waves beyond finite time intervals, we formulate the existence problem for traveling waves via spatial dynamics. There exist no spatially decaying (solitary) waves because of an infinite-dimensional center manifold in the spatial dynamics formulation. Existence of traveling modulating pulse solutions which are solitary waves with small oscillatory tails at very long distances from the pulse core is proven by using a local center-saddle manifold. We show that the variational formulation fails to capture existence of such modulating pulse solutions even in the singular limit of zero wave speeds where true (standing) solitary waves exist. Propagation of a traveling solitary wave and formation of a small oscillatory tail outside the pulse core is shown in numerical simulations of the NLS equation on the periodic graph.

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arXiv2024

Ground states on a fractured strip and one dimensional reduction

We consider the nonlinear Schr\''odinger equation on a strip with Neumann boundary conditions and a delta condition on the $x$-axis. First, we show the existence of ground states as minimizers of the action or of the energy under suitable constraints. Second, we prove that the energy minimizers converge to the ground state on the line with a delta condition as the amplitude of the strip shrinks to zero.

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arXiv2024

A minimal mass blow-up solution on a nonlinear quantum star graph

We construct a finite-time blow-up solution to the mass-critical focusing nonlinear Schrödinger equation on a metric star graph with an arbitrary number of edges. We show that all solutions are global if their mass is smaller than an explicit constant, called "minimal mass". We then construct a solution with minimal mass and arbitrary energy, which blows up in finite time at the vertex of the star graph. The blow-up profile and blow-up speed are explicitly characterized. The main novelty of the paper is the construction of the blow-up profile in time-dependent domains of singularly perturbed Laplacians.

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arXiv2024

Computation of excited states for the nonlinear Schr{ö}dinger equation: numerical and theoretical analysis

Our goal is to compute excited states for the nonlinear Schr{ö}dinger equation in the radial setting. We introduce a new technique based on the Nehari manifold approach and give a comparison with the classical shooting method. We observe that the Nehari method allows to accurately compute excited states on large domains but is relatively slow compared to the shooting method.

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arXiv2024

Variational properties of space-periodic standing waves of nonlinear Schr{ö}dinger equations with general nonlinearities

Periodic waves are standing wave solutions of nonlinear Schr\''odinger equations whose profile is periodic in space dimension one. We consider general nonlinearities and provide variational characterizations for the periodic wave profiles. This involves minimizing energy while keeping mass and momentum constant, as well as minimizing the action over the Nehari manifold. These variational approaches are considered both in the periodic and anti-periodic settings, and for focusing and defocusing nonlinearities. In appendix, we study the existence properties of periodic solutions of the triple power nonlinearity.

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arXiv2021

Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schr{ö}dinger equation

For the double power one dimensional nonlinear Schr{ö}dinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.

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arXiv2021

Gradient Flow Approach to the Calculation of Stationary States on Nonlinear Quantum Graphs

We introduce and implement a method to compute stationary states of nonlinear Schr\''odinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schr\''odinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.

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arXiv2021

Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schr{ö}dinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schr{ö}dinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schr{ö}dinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.

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arXiv2020

Stability of the multi-solitons of the modified Korteweg-de Vries equation

We establish the nonlinear stability of $N$-soliton solutions of the modified Korteweg-de Vries (mKdV) equation. The $N$-soliton solutions are global solutions of mKdV behaving at (positive and negative) time infinity as sums of $1$-solitons with speeds $0<c_1<\cdots< c_N$.The proof relies on the variational characterization of $N$-solitons. We show that the $N$-solitons realize the local minimum of the $(N+1)$-th mKdV conserved quantity subject to fixed constraints on the $N$ first conserved quantities.To this aim, we construct a functional for which $N$-solitons are critical points, we prove that the spectral properties of the linearization of this functional around a $N$-soliton are preserved on the extended timeline, and we analyze the spectrum at infinity of linearized operators around $1$-solitons. The main new ingredients in our analysis are a new operator identity based on a generalized Sylvester law of inertia and recursion operators for the mKdV equation.

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arXiv2016

On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation

We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we show the persistence of the stationary black soliton of the unperturbed problem as a solution. We also prove the existence of another branch of non-trivial stationary waves. Depending on the attractive or repulsive nature of the Dirac perturbation and of the type of stationary solutions, we prove orbital stability via a variational approach, or linear instability via a bifurcation argument.

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arXiv2016

Stability of periodic waves of 1D cubic nonlinear Schr{ö}dinger equations

We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schr{ö}dinger equations. First, we give global variational characterizations of each of these periodic waves, which in particular provide alternate proofs of their orbital stability with respect to same-period perturbations, restricted to certain subspaces. Second, we prove the spectral stability of the cnoidal waves against same-period perturbations (in a certain parameter range), and provide an alternate proof of this (known) fact for the snoidal waves, which does not rely on complete integrability. Third, we give a rigorous version of a formal asymptotic calculation of Rowlands to establish the instability of a class of real-valued periodic waves in 1D, which includes the cnoidal waves of the 1D cubic focusing nonlinear Schr{ö}dinger equation, against perturbations with period a large multiple of their fundamental period. Finally, we develop a numerical method to compute the minimizers of the energy with fixed mass and momentum constraints. Numerical experiments support and complete our analytical results.

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arXiv2016

Stability of multi-solitons for the derivative nonlinear Schr{ö}dinger equation

The nonlinear Schr{ö}dinger equation with derivative cubic nonlinearity admits a family of solitons, which are orbitally stable in the energy space. In this work, we prove the orbital stability of multi-solitons configurations in the energy space, under suitable assumptions on the speeds and frequencies of the composing solitons. The main ingredients of the proof are modulation theory, energy coercivity and monotonicity properties.

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arXiv2016

Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations

We will first review known results on multi-solitons of dispersive partial differential equations, which are special solutions behaving like the sum of many weakly-interacting solitary waves. We will then describe our recent joint work with Dong Li on nonlinear Schrödinger equations: Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of a soliton train which is a multi-soliton composed of infinitely many solitons. In the 1D case, we can add to the infinite train an additional half-kink, which is a solution with a non-zero background at minus infinity.

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arXiv2015

Multi-speed solitary waves of nonlinear Schr{ö}dinger systems: theoretical and numerical analysis

We consider a system of coupled nonlinear Schr{ö}dinger equations in one space dimension. First, we prove the existence of multi-speed solitary waves, i.e solutions to the system with each component behaving at large times as a solitary wave. Then, we investigate numerically the interaction of two solitary waves supported each on one component. Among the possible outcomes, we find elastic and inelastic interactions, collision with mass extraction and reflexion.

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