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Physics, Fluids & Plasmas
Alexey Tikan, Felicien Bonnefoy, Giacomo Roberti, Gennady El, Alexander Tovbis, Guillaume Ducrozet, Annette Cazaubiel, Gaurav Prabhudesai, Guillaume Michel, Francois Copie, Eric Falcon, Stephane Randoux, Pierre Suret
Summary: The Peregrine soliton (PS) is a prototype nonlinear structure that captures the properties of rogue waves. Recent research has shown that the PS can emerge independently of its solitonic content from partially radiative or solitonless initial data. In this study, the researchers controlled the occurrence of the PS in space-time by adjusting the initial chirp. The proposed method of nonlinear spectral engineering was found to be robust to higher-order nonlinear effects.
PHYSICAL REVIEW FLUIDS
(2022)
Article
Engineering, Mechanical
Jie Jin, Yi Zhang, Rusuo Ye, Lifei Wu
Summary: The coupled mixed derivative nonlinear Schrodinger equations, correlated with Lax pairs involving 3 x 3 matrices, have been studied and various types of solutions have been obtained using the Darboux transformation. The results have significant implications for understanding integrable systems in different physical contexts.
NONLINEAR DYNAMICS
(2023)
Article
Physics, Fluids & Plasmas
F. Demontis, G. Ortenzi, G. Roberti, M. Sommacal
Summary: This study investigates the behavior of the (1 + 1) focusing nonlinear Schrodinger equation under different initial conditions. The authors provide criteria for the occurrence of blowup or relaxation using self-similar solutions. They also explore the effects of dispersion on the formation of rogue waves, highlighting the role of the chirp in determining the prevailing scenario. The findings have potential implications for experiments in nonlinear optics and fluid dynamics.
Article
Mathematics, Applied
Mariana Haragus, Dmitry E. Pelinovsky
Summary: In this study, we discuss the linear instability of the Kuznetsov-Ma breathers and the Akhmediev breathers of the focusing nonlinear Schrodinger equation using tools from the theory of integrable systems. By constructing the exact solutions of the Lax system associated with the breathers using the Darboux transformation, we obtain a full description of the Lax spectra for the two breathers, including multiplicities of eigenvalues. Solutions of the linearized NLS equations are then obtained from the eigenfunctions and generalized eigenfunctions of the Lax system.
JOURNAL OF NONLINEAR SCIENCE
(2022)
Article
Mathematics, Applied
Jin-Jie Yang, Shou-Fu Tian, Zhi-Qiang Li
Summary: The Riemann-Hilbert problem is developed to study the focusing behavior of the nonlinear Schrodinger equation with multiple high-order poles. By employing Laurent expansion and solving an algebraic system, soliton solutions corresponding to the transmission coefficient are obtained.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics
Chenjie Fan
Summary: We studied the non-scattering L-2 solution u to the radial focusing mass-critical nonlinear Schrodinger equation with mass just above the ground state, and showed the existence of a time sequence {t(n)} where u(t(n)) weakly converges to the ground state Q up to scaling and phase transformation. We also provided some partial results on the mass concentration phenomena of the minimal mass blow-up solution.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Physics, Mathematical
Marco Bertola, Thomas Bothner
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2015)
Article
Physics, Multidisciplinary
Marco Bertola, Di Yang
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2015)
Article
Mathematics, Applied
Marco Bertola, Boris Dubrovin, Di Yang
PHYSICA D-NONLINEAR PHENOMENA
(2016)
Article
Physics, Mathematical
M. Bertola, M. Cafasso, V. Rubtsov
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2018)
Article
Mathematics, Applied
Ferenc Balogh, Marco Bertola, Seung-Yeop Lee, Kenneth D. T-R McLaughlin
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2015)
Article
Mathematics
Marco Bertola, Mattia Cafasso
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2015)
Article
Mathematics, Applied
M. Bertola, A. Tovbis
ANALYSIS AND MATHEMATICAL PHYSICS
(2015)
Article
Physics, Mathematical
M. Bertola, J. Harnad
JOURNAL OF MATHEMATICAL PHYSICS
(2019)
Article
Physics, Mathematical
M. Bertola, J. Harnad, B. Runov
JOURNAL OF MATHEMATICAL PHYSICS
(2020)
Article
Physics, Mathematical
M. Bertola, D. Korotkin
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2020)
Correction
Physics, Mathematical
M. Bertola
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Marco Bertola
Summary: The paper first introduces the concept of Pade approximation of Weyl-Stiltjes transforms on compact Riemann surfaces of higher genus, and characterizes these orthogonal sections through a matrix equation. It then extends this idea to explore its connection to integrable systems, demonstrating the relationship through defining pairing relationships and studying the properties of τ functions, showing its relevance to Krichever construction of algebro-geometric solutions.
ANALYSIS AND MATHEMATICAL PHYSICS
(2021)
Article
Physics, Mathematical
M. Bertola, D. Korotkin
Summary: This paper presents a new Hamiltonian formulation of Schlesinger equations using the dynamical r-matrix structure, showing the corresponding symplectic form as the pullback of a natural symplectic form under the monodromy map. It is demonstrated that Fock-Goncharov coordinates are log-canonical for the symplectic form, and the Jimbo-Miwa-Ueno τ-function is interpreted as the generating function of the monodromy map, resolving a recent conjecture by A. Its, O. Lisovyy, and A. Prokhorov.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics
M. Bertola
Summary: This study considers the notion of denominators in Pade-like approximation problems on a Riemann surface, which are related to the classical concept of orthogonality over a contour. It investigates a specific setup where the Riemann surface is a real elliptic curve with two components and the measure of orthogonality is supported on one of the real ovals. By characterizing the problem using a Riemann-Hilbert framework, the strong asymptotic behavior of the corresponding orthogonal functions for large degree is determined. This research highlights the influential role of vector bundles and the nonabelian Cauchy kernel in this simplified setting, indicating the challenges faced by the steepest descent method on a Riemann surface.
JOURNAL OF APPROXIMATION THEORY
(2022)
Article
Mathematics, Applied
M. Bertola, R. Jenkins, A. Tovbis
Summary: In this paper, we obtain Fredholm type formulas for partial degenerations of theta functions on nodal curves, focusing on those of genus one. We apply these formulas to study 'many-soliton' solutions on an elliptic background wave for the Korteweg-de Vries equation. The expressions for the solitary disturbances' speed and their interaction kernel are explicitly obtained in terms of Jacobi theta functions. We also show the convergence of genus N + 1 finite gap solutions to the deterministic cnoidal wave solution as the number of bands degenerate to a genus one nodal curve. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.
Article
Mathematics, Applied
Davide Parise, Alessandro Pigati, Daniel Stern
Summary: This paper studies the self-dual Yang-Mills-Higgs energies on a closed Riemannian manifold and proves their convergence to minimal submanifolds. The author establishes a connection between the energies and the Euler class by introducing a suitable gauge invariant Jacobian, and shows the existence of a recovery sequence under certain conditions. Furthermore, a comparison between the min-max values obtained from the Almgren-Pitts theory and the Yang-Mills-Higgs framework is made, with the former always providing a lower bound for the latter.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Wenkui Du, Robert Haslhofer
Summary: This paper explores ancient noncollapsed mean curvature flows and provides insights into their behavior and properties through spectral analysis and precise asymptotic analysis in various cases.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2024)