Special Session 71: Progress in Partial Differential Equations of Mathematical Physics: Theory and Methods

On the nonlinear Schr\odinger equation on the half line: analysis and numerical solution

Dimitra Antonopoulou National and Kapodistrian University of Athens Greece Co-Author(s):

Abstract: We consider the NLS equation with a cubic nonlinearity posed on the half line. In the defocusing case, we prove that if the Dirichlet data have a sufficient polynomial decay then the Neumann data also have a proper decay so that the Fokas method is applicable. Moreover, we prove that the solution in the $L^4$ space norm converges to $0$ as $t\rightarrow \infty$ establishing also the absence of solitons for this case. In the focusing case, decay of the Neumann data is proven (for properly decaying Dirichlet data), but only under the assumption of decay of the solution as $t \rightarrow \infty$. A Crank-Nicolson finite differences nonlinear scheme is introduced for the numerical solution of the problem on the positive semi-axis with experimental rates of convergence of order $2$. We present numerical simulations for the soliton propagation with various initial conditions. The results are joint with Spyros Kamvissis.

Extreme Superposition: Rogue Waves of Infinite Order, Universality, and Anomalous Temporal Decay

Deniz Bilman University of Cincinnati USA Co-Author(s):

Abstract: Focusing nonlinear Schr\"odinger equation serves as a universal model for the amplitude of a wave packet in a general one-dimensional weakly-nonlinear and strongly-dispersive setting that includes water waves and nonlinear optics as special cases. Rogue waves of infinite order are a novel family of solutions of the focusing nonlinear Schr\"odinger equation that emerge universally in a particular asymptotic regime involving a large-amplitude and near-field limit of a broad class of solutions of the same equation. In this talk, we will present several recent results on the emergence of these special solutions along with their interesting asymptotic and exact properties. Notably, these solutions exhibit anomalously slow temporal decay and are connected to the third Painlev\'e equation. Finally, we will extend the emergence of rogue waves of infinite order to the first several flows of the AKNS hierarchy--allowing for arbitrarily many simultaneous flows. Time permitting, we will report on recent work regarding their space-time asymptotic behavior under an arbitrary flow from the hierarchy.

Bore Propagation Models

Jerry L Bona University of Illinois at Chicago USA Co-Author(s):

Abstract: We sketch the development of models for bore propagation on rivers. We offer an analysis of one of the most comprehensive models.

The Unified Transform Method for variable-coefficient equations

Bernard Deconinck University of Washington USA Co-Author(s):    Matthew Farkas

Abstract: The Unified Transform Method or Method of Fokas has been successful in explicitly solving interface problems for constant coefficient linear evolution equations. Stated differently, the method can solve linear evolution equations with piecewise constant coefficients. Such problems arise in lots of applications. Even more important are linear evolution equations with spatially variable coefficients. I will show how by considering the limit of piecewise-constant coefficient problems, we may obtain explicit solution representations for variable coefficient problems.

Some remarks on the semiclassical asymptotics of eigenvalues and resonances for matrix-valued operators

Setsuro SF Fujiie RITSUMEIKAN UNIVERSITY (Department of Mathematical Scinence) Japan Co-Author(s):

Abstract: 1

Initial and boundary value problems for dispersive equations

Alex Himonas University of Notre Dame USA Co-Author(s):

Abstract: In this talk we discuss the solving of initial and boundary value problems for dispersive evolution equations. Models of such equations are the nonlinear Schr\odinger equation, the Korteweg-de Vries equation, and dispersive regularizations of Camassa-Holm type equations. For data in Sobolev spaces, we will present optimal well-posedness results based on sharp multilinear estimates motivated by estimating the solution of the forced linear problem in Bourgain solution spaces. Also, we shall present some implications of these estimates in the analytic theory of these equations.

New transform methods for doubly connected planar domains

Jesse Hulse University of Manitoba Canada Co-Author(s):

Abstract: A new transform-based technique that generalizes the Unified Transform Method is developed for bounded doubly connected domains as a novel way to numerically solve boundary value problems for holomorphic functions and solutions to the Laplacian. This work utilizes the Szeg\"{o} kernel for the annulus and generalizes methods for circular and simply connected domains developed by Crowdy (2015) and H., Lanzani, Llewellyn Smith, and Luca (2025). Time permitting, an application will be shown.

Models of soliton gas for the AKNS hierarchy

Robert Jenkins University of Central Florida USA Co-Author(s):

Abstract: In this talk I will present a collection of recent results on modeling soliton gas. Specifically, I describe how one can take an $N$-soliton solution of any equation in the AKNS hierarchy and pass in the $N \to \infty$ limit to a particular model of soliton gas by making certain reasonable assumptions on the asymptotic density of eigenvalues and the norming constants which define the $N$-soliton spectrally. Time permitting I will discuss some results on the large time evolution of such solutions,

Existence of traveling wave solutions for the nonlocal derivative nonlinear Schr\{o}dinger equation

Mukhtar Karazym Nazarbayev University Kazakhstan Co-Author(s):    Amin Esfahani, Adilbek Kairzhan

Abstract: We are interested in traveling wave solutions of the nonlocal derivative nonlinear Schr\{o}dinger (nonlocal DNLS) equation $$ i u_t - u_{xx} - b |u|^2 u + i \alpha |u|^2 u_x + i \beta u^2 \bar{u}_x + \gamma u\, \partial_x \mathcal{H}(|u|^2) = 0, $$ where $b, \alpha, \beta, \gamma \in \mathbb{R}$ are parameters, and $\mathcal{H}$ denotes the Hilbert transform $$ \mathcal{H}u(x,t) = \frac{1}{\pi}\,\mathrm{p.v.}\int_{\mathbb{R}} \frac{u(y,t)}{x-y}\,dy. $$ Depending on the sign of the parameters, we consider subcritical and critical minimization problems. Our main techniques are the Nehari manifold method and the concentration-compactness principle of P.L. Lions.

New transform methods for boundary value problems in planar domains

Elena Luca The Cyprus Institute Cyprus Co-Author(s):    Jesse J. Hulse (University of Manitoba), Loredana Lanzani (University of Bologna), Stefan G. Llewellyn Smith (University of California San Diego)

Abstract: The Unified Transform Method (UTM), introduced by A.S. Fokas in the late 1990s, is a method for analyzing boundary value problems for linear and integrable nonlinear PDEs. Since its inception, the UTM has attracted significant attention within the mathematics community and has been extended to address a wide range of problems. For Laplace's equation, Fokas and Kapaev (2003) developed a transform method for boundary value problems in simply-connected convex polygons. Their approach initially utilized various techniques, including spectral analysis of parameter-dependent ODEs and Riemann--Hilbert methods. Later, Crowdy (2015) showed that this method can be reformulated within a complex function-theoretic framework, leading to the construction of new transform pairs tailored to circular domains (domains bounded by circular arcs, with convex polygons as a special case). We present a new transform-based approach for boundary value problems for Laplace's equation in more general planar domains, specifically in simply connected Lipschitz domains (including non-convex domains). The key ingredient is the exploitation of the properties of the Szeg\H o kernel and its connection with the Cauchy kernel, which enables the construction of transform pairs for analytic functions in such domains. Several examples are presented, together with numerical implementations illustrating the effectiveness of the transform pairs.

Homogenization results for the nonlinear Schr\\odinger equation

Maria Ntekoume Concordia University Canada Co-Author(s):    Benjamin Harrop-Griffiths

Abstract: The cubic nonlinear Schr\odinger equation (NLS) arises as a model for the propagation of intense continuous wave laser beams in a homogeneous medium. In practice, it has also been successful in modeling optical experiments in inhomogeneous settings. This suggests the occurrence of homogenization, that is, in a large scale limit, solutions to the inhomogeneous equation converge to the solution of a homogeneous NLS. We will review some recent homogenization results for physically motivated examples. While materials with a periodic structure are the most natural setting to consider this problem, our examples include a model where strong defects are sprinkled randomly across the medium. Part of this talk is based on joint work with B. Harrop-Griffiths.

Uncovering bistability phenomena in two-layer Couette flow experiments using nonlocal evolution equations

Demetrios T Papageorgiou Imperial College London England Co-Author(s):    Xingyu Wang, Department of Mathematics, Imperial College London Pierre Germain, Faculty of Mathematics, Universitat Wien

Abstract: The stability of interfacial long waves in two-layer plane Couette flow is investigated using a nonlinear, nonlocal asymptotic model derived from the Navier-Stokes equations and valid for thin upper layers. Nonlocality enters through a coupling of the thin and main layers, and crucial inertial effects are retained. The models generically support bistability phenomena observed in experiments where two stable travelling waves, one unimodal and the other bimodal, are recorded at the same lid velocity. In direct comparisons with experiments, the models show remarkable agreement, both qualitatively and quantitatively. The two stable travelling waves are identified, and their basins of attraction characterised via large-time computations for different initial conditions. We also identify a new symmetry-breaking travelling-wave branch bifurcating from the bimodal family, compute higher-wavenumber travelling-wave branches, and present time-periodic orbits arising via Hopf bifurcations.

The surprising world of third-order dispersion

Beatrice Pelloni Heriot-Watt University Scotland Co-Author(s):

Abstract: I will describe the solution of third-order diseprsive equations when the initial condition has low regularity, for a variety of boundary conditions.

On the critical lengths and controllability of the Kawahara equation

Lionel ROSIER Universite du Littoral Cote d`Opale France Co-Author(s):    Fagner D. Araruna and Gleb G. Doronin

Abstract: This paper deals with the exact controllability of the Kawahara equation posed on a finite interval with one boundary control. We prove that the local exact controllability of Kawahara equation holds provided that the length of the interval does not belong to a countable set of critical lengths.

Scale invariant regularity estimates for the Neumann problem in Lipschitz domains

Georgios Sakellaris Aristotle University of Thessaloniki Greece Co-Author(s):

Abstract: We will discuss the Neumann Green function and scale invariant regularity estimates for the equation $-div(A\nabla u+bu)+c\nabla u+du=-div f+g$ with Neumann data in Lipschitz domains $\Omega\subseteq\mathbb R^n$. Under the assumption that $A$ is elliptic and bounded, we will see a necessary structural condition on the lower order coefficients that guarantees at most one dimensional kernels, as well as boundedness close to the boundary. Under the optimal assumptions $b,c\in L^n$ and $d\in L^{n/2}$, we will then show estimates for the $L^2$ theory that are scale invariant: that is, they depend only on the norms of the coefficients and the Lipschitz character of $\Omega$. One difficulty will be the existence of nontrivial kernels, which will differentiate the theorems considered, but this will be identified by a specific integral of the coefficient $d$. We will also discuss the analogue of Green`s function for the Neumann problem, called the Neumann Green function, and show that it satisfies scale invariant estimates in appropriate weak $L^p$ spaces. These estimates will lead to scale invariant local boundedness, under specific assumptions for the Neumann data in Lorentz spaces that are both necessary and optimal.

Higher order dispersive equations on the half-line

Fangchi Yan Virginia Tech USA Co-Author(s):    Alex Himonas

Abstract: In this talk, we study initial-boundary value problems for nonlinear dispersive equations with higher order dispersion on the half-line. Such problems arise naturally in physical models where wave propagation is influenced by both nonlinear effects and dispersion. Our approach is based on the Fokas method, which provides a systematic framework for solving the associated linear problems. Using this method, we establish optimal well-posedness results for a class of higher order nonlinear Schr\odinger and KdV-type equations.