Schrödinger type asymptotic model for wave propagation

Supervision

Raphaël Côte and Benjamin Mélinand (IRMA, Strasbourg)

Laboratory and team

IRMA, Strasbourg - Teams “MOCO" and "Analysis"

Subject description

In this thesis we are interested in the mathematical study of wave propagation through different asymptotic models with high frequency Schrödinger behavior. In a first axis, we want to rigorously derive and study a unidirectional asymptotic model of wave equations. This equation is written in the form √(1-∂_x^2 ) ∂_t u+ ∂_x u+ ∂_x^3 u+u ∂_x u=0, (t,x)∈R ×R. At low frequency, the symbol of the linear operator is of the Korteweg-de Vries (KdV) type, and at high frequency, of the Schrodinger type. As fas as we are aware, this model is the first to show this mixed behavior associating two of the most intensively studied dispersive equations. As a first step we plan to show that this equation approaches the wave equation with the same precision as the KdV or BBM equation. We will use for this the methodology described in [Lan13]. In a second step, we will focus on the local and global well-posedness of this equation, with particular attention to smoothness required on the initial data. This point is close to the work done on the BBM equation (see for example [BT09]). Next, we are to study the existence and stability of solitons (waves that propagate while retaining their shape, and which thus achieve an equilibrium between the dispersive and nonlinear effects). We have in mind to apply the techniques used in [KLPS25] and the associated references. Let us insist on two points. First, the Schrödinger-type high-frequency behavior is an important point for obtaining the existence of such objects: in fact, it boils down to a non-local elliptic equation. Second, a low regularity existence result is very important for studying stability. Finally, we also plan to study the behavior of these solitons according to their speed. At low speed, they are expected to behave like KdV solitons, possibly with monotonicity/Kato smoothing properties; and at high speeds, like those of the Schrodinger equation with quadratic nonlinearity. The goal is to make this intuition rigorous. To complete our study we can rely on numerical simulations (as was done for example in [KLPS25] on other equations). We typically have in mind to study numerically the long time behavior. The second line of research is to construct an asymptotic model satisfying the same properties as our scalar equation, but without the unidirectional character. It is therefore a question of deriving a system of equations (typically on the surface and the velocity of the fluid). We also want this système to be globally well posed, for initial data with low regularity. For this, we will start from the abcd Boussinesq systems, which were derived for the first time in [BCS02] and which will be modified to satisfy the desired properties. Once this is done, we will also study in this new framework the existence and stability of solitons. 

References
[BCS02] J. L. Bona, M. Chen, and J.-C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci., 12(4) :283–318, 2002.
[BT09] J. Bona and N/ Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete Contin. Dyn. Syst., 23(4) :1241–1252, 2009.
[KLPS25] C. Klein, F. Linares, D. Pilod, and J.-C. Saut. On the Benjamin and related equations. Bull. Braz. Math. Soc. (N.S.), 56(1) :Paper No. 4, 27, 2025.
[Lan13] D. Lannes. The water waves problem, volume 188 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics.

Related mathematical skills

M2 Analysis & PDEs

IRMIA++ is one of the 15 Interdisciplinary Thematic Institute (ITI) of the University of Strasbourg. It brings together a research cluster and a master-doctorate training program, relying on 12 research teams and 9 master tracks.

It encompasses all mathematicians at Université de Strasbourg, with partners in computer science and physics. ITI IRMIA++ builds on the internationally renowned research in mathematics in Strasbourg, and its well-established links with the socio-economic environment. It promotes interdisciplinary academic collaborations and industrial partnerships.

A core part of the IRMIA++ mission is to realize high-level training through integrated master-PhD tracks over 5 years, with common actions fostering an interdisciplinary culture, such as joint projects, new courses and workshops around mathematics and its interactions.

Selection will rely on the professional project of the candidate, his/her interest for interdisciplinarity and academic results.