arXiv subjects
Publications and source records attributed to Jean-Claude Saut.
In this work we consider the well posed version of the Kaup-Broer-Kuperschmidt system in two dimensions. We numerically construct soliton type solutions and show that they are unstable both against dispersion and singularity formation. Further, we study line solitons and their stability, as well as generally localised initial data. In either case we fail to find stable structures.
This paper considers a class of non-local equations that are weakly dispersive perturbations of the inviscid Burgers equation, which includes the Fornberg-Whitham equation as a special case. We precise the known results on finite time blow-up (shock formation) by constructing a blowup solution which displays a `shock-like' singularity (called wave breaking) at one single point. Moreover, this solution converges asymptotically in the self-similar variables to a stable self-similar solution of the inviscid Burgers equation, and also possesses a Hölder $C^{1/3}$ regularity at the blowup point.
This paper aims to give a refined wave breaking description of the Cauchy problem to the one-dimensional nonlinear shallow water equations providing a sharp estimate of the lifespan of the solutions depending on the amplitude and topography parameters, under a non-cavitation condition which excludes the scenario that the solutions have compact support. We construct smooth initial data with finite $\dot{H}^5$-norm such that the $L^\infty$-norm of the spatial derivative of the solution blows up at one single point in finite time with a precise blowup profile.
We prove the long time existence of solutions of a Boussinesq system with strong topography variations.
We prove the existence on long time scales of the solutions to the Cauchy problem for a version of weakly transverse Boussinesq systems arising in the modeling of surface water waves. This system is much more complicated than the isotropic Boussinesq systems because dispersion is only present in the x-direction, leading to anisotropic eigenvalues in the linearized system. This anisotropic character leads to loss of y-derivatives for the solutions. To overcome this main difficulty our strategy is to symmetrize the system by introducing suitable good unknowns in the sense of [3].
This article represents a first step toward understanding the long-time dynamics of solutions for the Intermediate Long Wave equation (ILW). While this problem is known to be both completely integrable and globally well-posed in $H^{\frac{3}{2}}$, much less seems to be known concerning its long-time dynamics. Here we prove well-posedness at much lower regularity, namely an $L^2$ global well-posedness result. Then we consider the case of small and localized data and show that the solutions disperse up to cubic timescale.
This paper aims to show that the Cauchy problem of the Burgers equation with a weakly dispersive perturbation involving the Bessel potential (generalization of the Fornberg-Whitham equation) can exhibit wave breaking for initial data with large slope. We also comment on the dispersive properties of the equation.
We present a detailed numerical study of the stability under periodic perturbations of line solitons of two-dimensional, generalized Zakharov-Kuznetsov equations with various power nonlinearities. In the $L^{2}$-subcritical case, in accordance with a theorem due to Yamazaki we find a critical speed, below which the line soliton is stable. For higher velocities, the numerical results indicate an instability against the formation of lumps, solitons localized in both spatial directions. In the $L^2$-critical and supercritical cases but subcritical for the 1D generalized Korteweg-de Vries equation), the line solitons are shown to be numerically stable for small velocities, and strongly unstable for large velocities, with a blow-up observed in finite time.
This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and nonlinear Schrödinger (modified fNLS) equations, with the dispersions |D|^α\partial_x and |D|^{α+1}, respectively. We prove the global existence of small solutions for both the Cauchy problems to the modified fKdV and fNLS equations, with a modified scattering which has a logarithmic phase correction. Our results cover the full range -1<α<1, α\neq 0 for both the modified fKdV and fNLS equations.
We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial zeroes. This new difficulty is solved by the use of "good unknowns " and of normal form techniques.
This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space $L^2(\mathbb R^2)$. This result is sharp in view of the fact that the flow map cannot be $C^3$ continuous below $L^2(\mathbb R^2)$. Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space $U^2_S$ and its dual $V^2_S $ of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in $H^s(\mathbb R^2)$, $s>0$. Our results extend to the finite depth version of the Dysthe equation.
We prove wave breaking (shock formation) for some Whitham-type equations which include the Burgers-Hilbert equation, the fractional Korteweg-de Vries equation, and the classical Whitham equation. The result seems to be new for the Burgers-Hilbert equation. In the other cases we provide simpler proofs than the known ones.
We prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev-Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev-Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev-Petviashvili is locally well-posed in $H^s(\mathbb R^2)$, for $s>\frac74$, in the capillary-gravity setting.
We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.
Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical strati-fication of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm-Liouville problem associated to the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so called Brunt-Vais{ä}l{ä} frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approximation (i.e. neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a non-linear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluids systems.
We show that the limit infimum, as time $\,t\,$ goes to infinity, of any uniformly bounded in time $H^{3/2+}\cap L^1$ solution to the Intermediate Long Wave equation converge to zero locally in an increasing-in-time region of space of order $\,t/\log(t)$. Also, for solutions with a mild $L^1$-norm growth in time is established that its limit infimum converge to zero, as time goes to infinity. This confirms the non existence of breathers and other solutions for the ILW model moving with a speed "slower" than a soliton. We also prove that in the far field linearly dominated region, the $L^2$ norm of the solution also converges to zero as time approaches infinity. In addition, we deduced several scenarios for which the initial value problem associated to the generalized Benjamin-Ono and the generalized Intermediate Long Wave equations cannot possess time periodic solutions (breathers). Finally, as it was previously demonstrated in solutions of the KdV and BO equations, we establish the following propagation of regularity result : if the datum $u_0\in H^{3/2+}(\mathbb R)\cap H^m((x_0,\infty))$, for some $\;x_0\in\mathbb R,\,m\in Z^+,\,m\geq 2$, then the corresponding solution $u(t,\cdot)$ of the Intermediate Long Wave equation belongs to $H^m(β,\infty)$, for any $t>0$ and $β\in\mathbb R$.
We establish the long time existence of solutions for the "Boussinesq-Full Dispersion" systems modeling the propagation of internal waves in a two-layer system. For the two-dimensional Hamiltonian case we prove the global existence of small solutions of the corresponding systems.
This paper is concerned with the one-dimensional version of a specific member of the (abcd) family of Boussinesq systems having the higher possible dispersion. We will establish two different long time existence results for the solutions of the Cauchy problem. The proofs involve normal form transformations suitably modified away from the zero set of the phases.