• Discrete transparent boundary conditions for the mixed KDV-BBM equation, Christophe Besse, Pascal Noble, and David Sanchez.

    In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data.


  • A Friendly Review of Absorbing Boundary Conditions and Perfectly Matched Layers for Classical and Relativistic Quantum Waves Equations, Xavier Antoine Emmanuel Lorin, and Qinglin Tang.

    The aim of this paper is to describe concisely the recent theoretical and numerical developments concerning absorbing boundary conditions and perfectly matched layers for solving classical and relativistic quantum waves problems. The equations considered in this paper are the Schrödinger, Klein-Gordon and Dirac equations.


  • On the well-posedness of a quasi-linear Korteweg-de Vries equation, Colin Mietka.

    The Korteweg-de Vries equation (KdV) and various generalized, most often semi-linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general Hamil-tonian structure. This paper presents a local in time well-posedness result, that is existence and uniqueness of a solution and its continuity with respect to the initial data. The proof is based on the derivation of energy estimates, the major interest being the method used to get them. The goal is to make use of the structural properties of the equation, namely the skew-symmetry of the leading order term, and then to control subprincipal terms using suitable gauges as introduced by Lim and Ponce (SIAM J. Math. Anal., 2002) and developed later by Kenig, Ponce and Vega (Invent. Math., 2004) and S. Benzoni-Gavage, R. Danchin and S. Descombes (Electron. J. Diff. Eq., 2006). The existence of a solution is obtained as a limit from regularized parabolic problems. Uniqueness and continuity with respect to the initial data are proven using a Bona-Smith regularization technique.


  • Lagrange--Schwarz Waveform Relaxation domain decomposition methods for linear and nonlinear quantum wave problems, Xavier Antoine and Emmanuel Lorin.

    A Schwarz Waveform Relaxation (SWR) algorithm is proposed to solve by Domain Decomposition Method (DDM) linear and nonlinear Schrödinger equations. The symbols of the transparent fractional transmission operators involved in Optimized Schwarz Waveform Relaxation (OSWR) algorithms are approximated by low order Lagrange polynomials to derive Lagrange-Schwarz Waveform Relaxation (LSWR) algorithms based on local transmission operators. The LSWR methods are numerically shown to be computationally efficient, leading to convergence rates almost similar to OSWR techniques.


  • Frozen Gaussian Approximation based domain decomposition methods for the linear Schr{\"o}dinger equation beyond the semi-classical regime, Emmanuel Lorin, Xu Yang and Xavier Antoine.

    The paper is devoted to develop efficient domain decomposition methods for the linear Schrödinger equation beyond the semiclassical regime, which does not carry a small enough rescaled Planck constant for asymptotic methods (e.g. geometric optics) to produce a good accuracy, but which is too computationally expensive if direct methods (e.g. finite difference) are applied. This belongs to the category of computing middle-frequency wave propagation, where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy. Motivated by recent works of the authors on absorbing boundary conditions [X. Antoine et al, J. Comput. Phys., 277 (2014), 268–304] and [X. Yang and J. Zhang, SIAM J. Numer. Anal., 52 (2014), 808–831], we introduce Semiclassical Schwarz Waveform Relaxation methods (SSWR), which are seamless integrations of semiclassical approximation to Schwarz Waveform Relaxation methods. Two versions are proposed respectively based on Herman-Kluk propagation and geometric optics, and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods.


  • Global well-posedness of the Euler-Korteweg system for small irrotational data, Corentin Audiard and Boris Haspot.

    The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension d ≥ 3 for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if d ≥ 5, and a careful study of the nonlinear structure of the quadratic terms in dimension 3 and 4 involving the theory of space time resonance.


  • Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose-Einstein condensates, Christophe Besse, and Feng Xing.

    In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrödinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.


  • On the dynamics of floating structures, David Lannes.

    This paper addresses the floating body problem which consists in studying the interaction of surface water waves with a floating body. We propose a new formulation of the water waves problem that can easily be generalized in order to take into account the presence of a floating body. The resulting equations have a compressible-incompressible structure in which the interior pressure exerted by the fluid on the floating body is a Lagrange multi-plier that can be determined through the resolution of a d-dimensional elliptic equation, where d is the horizontal dimension. In the case where the object is freely floating, we decompose the hydrodynamic force and torque exerted by the fluid on the solid in order to exhibit an added mass effect; in the one dimensional case d = 1, the computations can be carried out explicitly. We also show that this approach in which the interior pressure appears as a Lagrange multiplier can be implemented on reduced asymptotic models such as the nonlinear shallow water equations and the Boussinesq equations; we also show that it can be transposed to the discrete version of these reduced models and propose simple numerical schemes in the one dimensional case. We finally present several numerical computations based on these numerical schemes; in order to validate these computations we exhibit explicit solutions in some particular configurations such as the return to equilibrium problem in which an object is dropped from a non-equilibrium position in a fluid which is initially at rest.


  • Counting unstable eigenvalues in Hamiltonian systems via commuting operators, Mariana Haragus, and Dmitry Pelinovsky.


  • Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis, Jean-François Coulombel.

    The aim of this article is to propose a systematic study of transparent boundary conditions for finite difference approximations of evolution equations. We try to keep the discussion at the highest level of generality in order to apply the theory to the broadest class of problems. We deal with two main issues. We first derive transparent numerical boundary conditions, that is, we exhibit the relations satisfied by the solution to the pure Cauchy problem when the initial condition vanishes outside of some domain. Our derivation encompasses discretized transport, diffusion and dispersive equations with arbitrarily wide stencils. The second issue is to prove sharp stability estimates for the initial boundary value problem obtained by enforcing the boundary conditions derived in the first step. We focus here on discretized transport equations. Under the assumption that the numerical boundary is non-characteristic, our main result characterizes the class of numerical schemes for which the corresponding transparent boundary conditions satisfy the so-called Uniform Kreiss-Lopatinskii Condition introduced in [GKS72]. Adapting some previous works to the non-local boundary conditions considered here, our analysis culminates in the derivation of trace and semigroup estimates for such transparent numerical boundary conditions. Several examples and possible extensions are given.


  • Artificial boundary conditions for the linearized Benjamin-Bona-Mahony equation, Christophe Besse, Benoît Mésognon-Gireau, and Pascal Noble.

    We consider various approximations of artificial boundary conditions for linearized Benjamin-Bona-Mahoney equation. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our transparent boundary conditions.


  • A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes, Arnaud Duran and Fabien Marche.

    In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.


  • Multilevel preconditioning techniques for Schwarz waveform relaxation domain decomposition methods for real-and imaginary-time nonlinear Schrödinger equations, Xavier Antoine and Emmanuel Lorin.

    This paper is dedicated to the derivation of a multilevel Schwarz Waveform Relaxation (SWR) Domain Decomposition Method (DDM) in real-and imaginary-time for the NonLinear Schrödinger Equation (NLSE). In imaginary-time, it is shown that the use of the multilevel SWR-DDM accelerates the convergence compared to the one-level SWR-DDM, resulting in an important reduction of the computational time and memory storage. In real-time, the method requires in addition the storage of the solution in overlapping zones at any time, but on coarser discretization levels. The method is numerically validated on the Classical SWR and Robin-based SWR methods but can however be applied to any SWR approach.


  • Numerical study of the stability of the Peregrine breather, Christian Klein and Mariana Haragus.

    The Peregrine breather is widely discussed as a model for rogue waves in deep water. We present here a detailed numerical study of perturbations of the Peregrine breather as a solution to the nonlinear Schr\"odinger (NLS) equations. We first address the modulational instability of the constant modulus solution to NLS. Then we study numerically localized and nonlocalized perturbations of the Peregrine breather in the linear and fully nonlinear setting. It is shown that the solution is unstable against all considered perturbations.


  • Transverse instability of periodic and generalized solitary waves for a fifth-order KP model, Mariana Haragus and Erik Wahl{\'e}n.

    We consider a fifth-order Kadomtsev-Peviashvili equation which arises as a two-dimensional model in the classical water-wave problem. This equation possesses a family of generalized line solitary waves which decay exponentially to periodic waves at infinity. We prove that these solitary waves are transversely spectrally unstable and that this instability is induced by the transverse instability of the periodic tails. We rely upon a detailed spectral analysis of some suitably chosen linear operators.


  • Discrete Artificial Boundary Conditions for the Korteweg-de Vries Equation, Christophe Besse, Matthias Ehrhardt, and Ingrid Lacroix-Violet.

    In this paper we consider two numerical scheme based on trapezoidal rule in time for the linearized KdV equation in one space dimension. The goal is to derive some suitable artificial boundary conditions for these two full discretization using Z-transformation. We give some numerical benchmark examples from the literature to illustrate our findings.


  • Long wave asymptotics for the Euler–Korteweg system, Sylvie Benzoni-Gavage and David Chiron.

    The Euler–Korteweg system (EK) is a fairly general nonlinear waves model in mathematical physics that includes in particular the fluid formulation of the NonLinear Schrödinger equation (NLS). Several asymptotic regimes can be considered, regarding the length and the amplitude of waves. The first one is the free wave regime, which yields long acoustic waves of small amplitude. The other regimes describe a single wave or two counter propagating waves emerging from the wave regime. It is shown that in one space dimension those waves are governed either by inviscid Burgers or by Korteweg-de Vries equations, depending on the spatio-temporal and amplitude scalings. In higher dimensions, those waves are found to solve Kadomtsev-Petviashvili equations. Error bounds are provided in all cases. These results extend earlier work on defocussing (NLS) (and more specifically the Gross–Pitaevskii equation), and sheds light on the qualitative behavior of solutions to (EK), which is a highly nonlinear system of PDEs that is much less understood in general than (NLS).


  • Amplitude equations for weakly nonlinear surface waves in variational problems, Sylvie Benzoni-Gavage and Jean-François Coulombel.

    Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle 'generically' admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and co-workers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012] that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011]. Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009].


  • Modelling turbulence generation in solitary waves on shear shallow water flows, Gaël Richard and Sergey Gavrilyuk.

    We derive a dispersive model of shear shallow water flows which takes into account a non-uniform horizontal velocity. This model generalizes the Green–Naghdi model to the case of shear flows. Besides the classical dispersion term in the Green–Naghdi model related to the acceleration of the free surface, it also contains a new dispersion parameter related to the flow structure. This parameter is related to the second moment of the velocity fluctuation with respect to the vertical coordinate. The distinction between shearing and turbulence based on the scale of variation of the velocity fluctuation is proposed. In particular, an equation for the turbulence generation is derived. Solitary waves for this model are obtained in explicit form. Comparison of solitary wave profiles with experimental ones is also performed. The agreement is very good apart from the small region near the top of the wave. (J. Fluid Mechanics Vol. 773 (2015), pp 49- 74)

  • On the boundary value problem for the Schrödinger equation: compatibility conditions and global existence, Corentin Audiard.

    We consider linear and nonlinear Schrödinger equations on a domain Ω with nonzero Dirichlet boundary conditions and initial data. We first study the linear boundary value problem with boundary data of optimal regularity (in anisotropic Sobolev spaces) with respect to the initial data. We prove well-posedness under natural compatibility conditions. This is essential for the second part, where we prove the existence and uniqueness of maximal solutions for nonlinear Schrödinger equations. Despite the nonconservation of energy, we also obtain global existence in several (defocusing) cases.

    pdf (Analysis & PDE Vol. 8 (2015), No. 5, 1113–1143)

  • Nonlinear wave-current interactions in shallow water, David Lannes and Fabien Marche.

    We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [9] that the motion of the waves could be described using an extended Green-Naghdi system. In this paper we propose an analysis of these equations, and show that they can be used to get some new insight into wave-current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We also propose a robust and simple numerical scheme validated on several examples. Finally, we give some examples of wave-current interactions with a non trivial vorticity field and topography effects.


  • The Leray-Gårding method for finite difference schemes, Jean-François Coulombel.

    In [Ler53] and [Går56], Leray and Gårding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in [Ler53, Går56 ] provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multiplier is the starting point of the argument by Rauch in [Rau72] for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels, and encompasses a somehow magical trick that has been known for a long time for the leapfrog scheme. More importantly, the existence and properties of the local multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems, which answers a problem raised by Trefethen, Kreiss and Wu [Tre84, KW93].


  • Co-periodic stability of periodic waves in some Hamiltonian PDEs, Sylvie Benzoni-Gavage, Colin Mietka, and Miguel Rodrigues.

    The stability of periodic traveling wave solutions to dispersive PDEs with respect to `arbitrary' perturbations is still widely open. The focus is put here on stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg--de Vries equation (qKdV), and the Euler--Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability, and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. Regarding solitary waves, the celebrated Grillakis--Shatah--Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, the most striking results obtained here can be summarized as: an odd value for the difference between N -- the size of the PDE system -- and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Since these stability criteria are merely encoded by the negative signature of matrices, they can at least be checked numerically. Various numerical experiments are presented, which clearly discriminate between stable cases and unstable cases for (qKdV), (EKE) and (EKL).


  • Spectral validation of the Whitham equations for periodic waves of lattice dynamical systems, Bugra Kabil and Miguel Rodrigues.

    In the present contribution we investigate some features of dynamical lattice systems near periodic traveling waves. First, following the formal averaging method of Whitham, we derive modulation systems expected to drive at main order the time evolution of slowly modulated wavetrains. Then, for waves whose period is commensurable to the lattice, we prove that the formally-derived first-order averaged system must be at least weakly hyperbolic if the background waves are to be spectrally stable, and, when weak hyperbolicity is met, the characteristic velocities of the modulation system provide group velocities of the original system. Historically, for dynamical evolutions obeying partial differential equations, this has been proved, according to increasing level of algebraic complexity, first for systems of reaction-diffusion type, then for generic systems of balance laws, at last for Hamiltonian systems. Here, for their semi-discrete counterparts, we give at once simultaneous proofs for all these cases. Our main analytical tool is the discrete Bloch transform, a discrete analogous to the continuous Bloch transform. Nevertheless , we needed to overcome the absence of genuine space-translation invariance, a key ingredient of continuous analyses.


  • Transverse dynamics of two-dimensional gravity-capillary periodic water waves, Mariana Haragus.

    We study the transverse dynamics of two-dimensional gravity–capillary periodic water waves in the case of large surface tension. In this parameter regime, predictions based on model equations suggest that periodic traveling waves are stable with respect to two-dimensional perturbations, and unstable with respect to three-dimensional perturbations which are periodic in the direction transverse to the direction of propagation. In this paper, we confirm the second prediction. We show that, as solutions of the full water-wave equations, the periodic traveling waves are linearly unstable under such three-dimensional perturbations. In addition, we study the nonlinear bifurcation problem near these transversely unstable two-dimensional periodic waves. We show that a one-parameter family of three-dimensional doubly periodic waves is generated in a dimension-breaking bifurcation. The key step of this approach is the analysis of the purely imaginary spectrum of the linear operator obtained by linearizing the water-wave equations at a periodic traveling wave. Transverse linear instability is then obtained by a perturbation argument for linear operators, and the nonlinear bifurcation problem is studied with the help of a center-manifold reduction and the classical Lyapunov center theorem.


  • The Kelvin-Helmholtz instabilities in two-fluids shallow water models, David Lannes and Mei Ming.

    The goal of this paper is to describe the formation of Kelvin-Helmholtz instabilities at the interface of two fluids of different densities and the ability of vari-ous shallow water models to reproduce correctly the formation of these instabilities. Working first in the so called rigid lid case, we derive by a simple linear anal-ysis an explicit condition for the stability of the low frequency modes of the inter-face perturbation, an expression for the critical wave number above which Kelvin-Helmholtz instabilities appear, and a condition for the stability of all modes when surface tension is present. Similar conditions are derived for several shallow water asymptotic models and compared with the values obtained for the full Euler equa-tions. Noting the inability of these models to reproduce correctly the scenario of formation of Kelvin-Helmholtz instabilities, we derive new models that provide a perfect matching. A comparisons with experimental data is also provided. Moreover, we briefly discuss the more complex case where the rigid lid is re-placed by a free surface. In this configuration, it appears that some frequency modes are stable when the velocity jump at the interface is large enough; we explain why such stable modes do not appear in the rigid lid case.


  • Well-posedness and shallow-water stability for a new Hamiltonian formulation of the water wave equations with vorticity , Angel Castro and David Lannes.

    In this paper we derive a new formulation of the water waves equa-tions with vorticity that generalizes the well-known Zakharov-Craig-Sulem for-mulation used in the irrotational case. We prove the local well-posedness of this formulation, and show that it is formally Hamiltonian. This new formu-lation is cast in Eulerian variables, and in finite depth; we show that it can be used to provide uniform bounds on the lifespan and on the norms of the solutions in the singular shallow water regime. As an application to these re-sults, we derive and provide the first rigorous justification of a shallow water model for water waves in presence of vorticity; we show in particular that a third equation must be added to the standard model to recover the velocity at the surface from the averaged velocity. The estimates of the present paper also justify the formal computations of [15] where higher order shallow water models with vorticity (of Green-Naghdi type) are derived.


  • Fully nonlinear long-waves models in presence of vorticity, Angel Castro and David Lannes.

    We study here Green-Naghdi type equations (also called fully nonlinear Boussinesq, or Serre equations) modeling the propagation of large amplitude waves in shallow water without smallness assumption on the amplitude of the waves. The novelty here is that we allow for a general vorticity, hereby allowing complex interactions between surface waves and currents. We show that the a priori 2+1-dimensional dynamics of the vorticity can be reduced to a finite cascade of two-dimensional equations: with a mechanism reminiscent of turbulence theory, vorticity effects contribute to the averaged momentum equation through a Reynolds-like tensor that can be determined by a cascade of equations. Closure is obtained at the precision of the model at the second order of this cascade. We also show how to reconstruct the velocity field in the 2 + 1 dimensional fluid domain from this set of 2-dimensional equations and exhibit transfer mechanisms between the horizontal and vertical components of the vorticity, thus opening perspectives for the study of rip currents for instance.


  • Variants of the focusing NLS equation. Derivation, justification and open problems related to filamentation, Éric Dumas and David Lannes and Jérémie Szeftel.

    The focusing cubic NLS is a canonical model for the propagation of laser beams. In dimensions 2 and 3, it is known that a large class of initial data leads to finite time blow-up. Now, physical experiments suggest that this blow-up does not always occur. This might be explained by the fact that some physical phenomena neglected by the standard NLS model become relevant at large intensities of the beam. Many ad hoc variants of the focusing NLS equation have been proposed to capture such effects. In this paper, we derive some of these variants from Maxwell's equations and propose some new ones. We also provide rigorous error estimates for all the models considered. Finally, we discuss some open problems related to these modified NLS equations.


  • A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations, David Lannes and Fabien Marche.

    We introduce a new class of two-dimensional fully nonlinear and weakly dispersive Green-Naghdi equations over varying topography. These new Green-Naghdi systems share the same order of precision as the standard one but have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces. For these new models, we develop a high order, well balanced, and robust numerical code relying on an hybrid finite volume and finite difference splitting approach. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a finite difference approach. Higher order accuracy in space and time is achieved through WENO reconstruction methods and through a SSP-RK time stepping. Particular effort is made to ensure positivity of the water depth. Numerical validations are then performed, involving one and two dimensional cases and showing the ability of the resulting numerical model to handle waves propagation and transformation, wetting and drying; some simple treatments of wave breaking are also included. The resulting numerical code is particularly efficient from a computational point of view and very robust; it can therefore be used to handle complex two dimensional configurations.


  • Stability of finite difference schemes for hyperbolic initial boundary value problems: numerical boundary layers, Jean-François Coulombel and Benjamin Boutin.

    In this article, we give a unified theory for constructing boundary layer expansions for dis-cretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels, while semigroup estimates were restricted, up to now, to numerical schemes with two time levels only.


  • Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function, Frédéric Chardard and Thomas Bridges.

    Partial differential equations in one space dimension and time, which are gradient-like in time with Hamiltonian steady part, are considered. The interest is in the case where the steady equation has a homoclinic orbit, representing a solitary wave. Such homoclinic orbits have two important geometric invariants: a Maslov index and a Lazutkin–Treschev invariant. A new relation between the two has been discovered and is moreover linked to transversal construction of homoclinic orbits: the sign of the Lazutkin–Treschev invariant determines the parity of the Maslov index. A key tool is the geometry of Lagrangian planes. All this geometry feeds into linearization about the homoclinic orbit in the time-dependent system, which is studied using the Evans function. A new formula for the symplectification of the Evans function is presented, and it is proven that the derivative of the Evans function is proportional to the Lazutkin–Treschev invariant. A corollary is that the Evans function has a simple zero if, and only if, the homoclinic orbit of the steady problem is transversely constructed. Examples from the theory of gradient reaction–diffusion equations and pattern formation are presented.


  • The Mach stem equation and amplification in strongly nonlinear geometric optics, Jean-François Coulombel and Mark Williams.

    We study highly oscillating solutions to a class of weakly well-posed hyperbolic initial boundary value problems. Weak well-posedness is associated with an amplification phenomenon of oscillating waves on the boundary. In the previous works [CGW14, CW14], we have rigorously justified a weakly nonlinear regime for semilinear problems. In that case, the forcing term on the boundary has amplitude O(ε^2) and oscillates at a frequency O(1/ε). The corresponding exact solution, which has been shown to exist on a time interval that is independent of ε ∈ (0,1], has amplitude O(ε). In this paper, we deal with the exact same scaling, namely O(ε^2) forcing term on the boundary and O(ε) solution, for quasilinear problems. In analogy with [CGM03], this corresponds to a strongly nonlinear regime, and our main result proves solvability for the corresponding WKB cascade of equations, which yields existence of approximate solutions on a time interval that is independent of ε ∈ (0,1]. Existence of exact solutions close to approximate ones is a stability issue which, as shown in [CGM03], highly depends on the hyperbolic system and on the boundary conditions; we do not address that question here. This work encompasses previous formal expansions in the case of weakly stable shock waves [MR83] and two-dimensional compressible vortex sheets [AM87]. In particular, we prove well-posedness for the leading amplitude equation (the "Mach stem equation") of [MR83] and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms. The latter case is solved under a crucial nonresonant assumption and a small divisor condition.


  • Fully discrete hyperbolic initial boundary value problems with nonzero initial data, Jean-François Coulombel.

    The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in ℓ 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b].


  • From Gross-Pitaevskii equation to Euler Korteweg system, existence of global strong solutions with small irrotational initial data, Corentin Audiard and Boris Haspot.

    In this paper we prove the global well-posedness for small data for the Euler Korteweg system in dimension N ≥ 3, also called compressible Euler system with quantum pressure. It is formally equivalent to the Gross-Pitaevskii equation through the Madelung transform. The main feature is that our solutions have no vacuum for all time. Our construction uses in a crucial way some deep results on the scattering of the Gross-Pitaevskii equation due to Gustafson, Nakanishi and Tsai in [28, 29, 30]. An important part of the paper is devoted to explain the main technical issues of the scattering in [29] and we give a detailed proof in order to make it more accessible. Bounds for long and short times are treated with special care so that the existence of solutions does not require smallness of the initial data in H s , s > N 2 . The optimality of our assumptions is also discussed.


  • Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations, Arnaud Duran and Fabien Marche.

    We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilize the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.


  • A Kinematic Conservation Law in Free Surface Flow, Sergey Gavrilyuk, H. Kalisch, and Z. Khorsand.

    The Green-Naghdi system is used to model highly nonlinear weakly dispersive wave propagating at the surface of a shallow layer of a perfect fluid. The system has three associated conservation laws which describe the conservation of mass, momentum, and energy due to the surface wave motion. In addition, the system features a fourth conservation law which is the main focus of this note. It will be shown how this fourth conservation law can be interpreted in terms of a concrete kinematic quantity connected to the evolution of the tangent velocity at the free surface. The equation for the tangent velocity is first derived for the full Euler equations in both two and three dimensional flows, and in both cases, it gives rise to an approximate balance law in the Green-Naghdi approximation which turns out to be identical to the fourth conservation law for this system.

  • Domain Decomposition Methods and High-Order Absorbing Boundary Conditions for the Numerical Simulation of the Time Dependent Schrödinger Equation with Ionization and Recombination by Intense Electric Field, Xavier Antoine, E. Lorin and A. D. Bandrauk.

    This paper is devoted to the efficient computation of the Time Dependent Schrödinger Equation (TDSE) for quantum particles subject to intense electromagnetic fields including ionization and recombination of electrons with their parent ion. The proposed approach is based on a domain decomposition technique, allowing a fine computation of the wavefunction in the vicinity of the nuclei located in a domain Ω1 and a fast computation in a roughly meshed domain Ω2 far from the nuclei where the electrons are assumed free. The key ingredients in the method are i) well designed transmission boundary conditions on ∂Ω1 (resp. ∂Ω2) in order to estimate the part of the wavefunction “leaving” Domain Ω1 (resp. Ω2), ii) a Schwarz waveform relaxation algorithm to accurately reconstruct the solution. The developed method makes it possible for electrons to travel from one domain to another without loosing accuracy, when the frontier or the overlapping region between two domains is crossed by the wavefunction.

  • Stability of periodic waves in Hamiltonian PDEs, Sylvie Benzoni-Gavage, Pascal Noble, and Miguel Rodrigues.

    Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability ~-~in fact, \emph{orbital} stability, since we are dealing with space-invariant problems~-~, is far from being straightforward when the best spectral stability we can expect is a \emph{neutral} one. Indeed, because of the Hamiltonian structure, the spectrum of the linearized equations cannot be bounded away from the imaginary axis, even if we manage to deal with the point zero, which is always present because of space invariance. Some other means make a crucial use of the underlying structure. This is clearly the case for the variational approach, which basically uses the Hamiltonian -~or more precisely, a constrained functional associated with the Hamiltonian and with other conserved quantities~- as a Lyapunov function. When it works, it is very powerful, since it gives a straight path to orbital stability. An alternative is the modulational approach, following the ideas developed by Whitham almost fifty years ago. The main purpose here is to point out a few results, for KdV-like equations and systems, that make the connection between these three approaches: spectral, variational, and modulational.


  • Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids, Sylvie Benzoni-Gavage, Pascal Noble, and Miguel Rodrigues.

    Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction-diffusion systems were first considered by Doelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], and viscous systems of conservation laws have been addressed by Johnson, Noble, Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipative models are considered, and a most basic question is investigated, namely the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg--de Vries equation, and the less known Euler--Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water waves theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre [Comm. Partial Differential Equations 2005], Oh and Zumbrun [Arch. Ration. Mech. Anal. 2003], and Johnson, Zumbrun and Bronski [Phys. D 2010]. In addition, reduced necessary conditions are obtained in the small amplitude limit. Then numerical investigations are carried out for the modulated equations of the Euler--Korteweg system with two types of 'pressure' laws, namely the quadratic law of shallow water equations, and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.