Solvable Nonlinear Evolution PDEs in Multidimensional Space

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schr\"odinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type. Isochronous variants of these evolution PDEs are also considered.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Heisenberg ferromagnetism as an evolution of a spherical indicatrix: localized solutions and elliptic dispersionless reduction

A geometrical formulation of Heisenberg ferromagnetism as an evolution of a curve on the unit sphere in terms of intrinsic variables is provided and investigated. Given a vortex filament moving in an incompressible Euler fluid with constant density (under the local induction approximation hypotheses), the solutions of the classical Heisenberg ferromagnet equation are represented by the corresponding spherical (or tangent) indicatrix. The equations for the time evolution of the indicatrix on the unit sphere are given explicitly in terms of two intrinsic variables, the geodesic curvature and the arc-length of the curve. Notably, by considering the evolution with respect to slow variables and neglecting the dispersive terms, a novel elliptic dispersionless reduction of the Heisenberg ferromagnet model is obtained. The length of the spherical indicatrix is proved not to be conserved. Finally, a totally explicit algorithm is provided, allowing to construct a solution of the Heisenberg ferromagnet equation from a solution of Nonlinear Schrodinger equation, and, remarkably, viceversa, allowing to construct a solution of Nonlinear Schrodinger equation from a solution of the Heisenberg ferromagnet equation. As expected from the Zakharov-Takhtajan gauge equivalence, in the reflectionless case such a two-way map between solutions is shown to preserve the Inverse Scattering Transform spectra, and thus the localization

AN INVERTIBLE TRANSFORMATION AND SOME OF ITS APPLICATIONS

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained is revealed. Various contexts are considered: algebraic and Diophantine equations, nonlinear Sturm–Liouville problems, dynamical systems (with continuous and with discrete time), nonlinear partial differential equations, analytical geometry, functional equations. While this transformation, in one or another context, is certainly known to many, it does not seem to be as universally known as it deserves to be, for instance it is not routinely taught in basic University courses (to the best of our knowledge). The main purpose of this paper is to bring about a change in this respect; but we also hope that some of the findings reported herein — and the multitude of analogous findings easily obtainable via this te..

Integrability and linear stability of nonlinear waves

It is well known that the linear stability of solutions of (Formula presented.) partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general (Formula presented.) matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for (Formula presented.) for the particular system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants

The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory

We develop the direct and inverse scattering theory of the linear eigenvalue problem associated with the classical Heisenberg continuous equation with in-plane asymptotic conditions. In particular, analyticity of the scattering eigenfunctions and scattering data, and their asymptotic behaviours are derived. The inverse problem is formulated in terms of Marchenko equations, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided

A new integrable model of long wave–short wave interaction and linear stability spectra

We consider the propagation of short waves which generate waves of much longer (infinite) wavelength. Model equations of such long wave–short wave (LS) resonant interaction, including integrable ones, are well known and have received much attention because of their appearance in various physical contexts, particularly fluid dynamics and plasma physics. Here we introduce a new LS integrable model which generalizes those first proposed by Yajima and Oikawa and by Newell. By means of its associated Lax pair, we carry out the linear stability analysis of its continuous wave solutions by introducing the stability spectrum as an algebraic curve in the complex plane. This is done starting from the construction of the eigenfunctions of the linearized LS model equations. The geometrical features of this spectrum are related to the stability/instability properties of the solution under scrutiny. Stability spectra for the plane wave solutions are fully classified in the parameter space together with types of modulational instabilities

Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation

The non-topological, stationary and propagating, soliton solutions of the classical continuous Heisenberg ferromagnet equation are investigated. A general, rigorous formulation of the Inverse Scattering Transform for this equation is presented, under less restrictive conditions than the Schwartz class hypotheses and naturally incorporating the non-topological character of the solutions. Such formulation is based on a new triangular representation for the Jost solutions, which in turn allows an immediate computation of the asymptotic behaviour of the scattering data for large values of the spectral parameter, consistently improving on the existing theory. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions (including breather-like and multipoles), and allowing their classification and descriptio

Aortic stenosis post-COVID-19: a mathematical model on waiting lists and mortality

Objectives To provide estimates for how different treatment pathways for the management of severe aortic stenosis (AS) may affect National Health Service (NHS) England waiting list duration and associated mortality. Design We constructed a mathematical model of the excess waiting list and found the closed-form analytic solution to that model. From published data, we calculated estimates for how the strategies listed under Interventions may affect the time to clear the backlog of patients waiting for treatment and the associated waiting list mortality. Setting The NHS in England. Participants Estimated patients with AS in England. Interventions (1) Increasing the capacity for the treatment of severe AS, (2) converting proportions of cases from surgery to transcatheter aortic valve implantation and (3) a combination of these two. Results In a capacitated system, clearing the backlog by returning to pre-COVID-19 capacity is not possible. A conversion rate of 50% would clear the backlog within 666 (533–848) days with 1419 (597–2189) deaths while waiting during this time. A 20% capacity increase would require 535 (434–666) days, with an associated mortality of 1172 (466–1859). A combination of converting 40% cases and increasing capacity by 20% would clear the backlog within a year (343 (281–410) days) with 784 (292–1324) deaths while awaiting treatment. Conclusion A strategy change to the management of severe AS is required to reduce the NHS backlog and waiting list deaths during the post-COVID-19 ‘recovery’ period. However, plausible adaptations will still incur a substantial wait to treatment and many hundreds dying while waiting

Rogue Wave Type Solutions and Spectra of Coupled Nonlinear Schrödinger Equations

The formation of rogue oceanic waves may be the result of different causes. Various factors (winds, currents, dispersive focussing, depth, nonlinear focussing and instability) make this subject intriguing, and yet its understanding is quite relevant to practical issues. Here, we deal only with the nonlinear character of this dynamics, which has been recognised as the main ingredient to rogue wave formation. In this perspective, the formation of rogue waves requires a non-vanishing and unstable background such as a nonlinear regular wave train with attractive self-interaction. The simplest, best known model of such dynamics is the universal nonlinear Schrödinger equation. This has proven to serve as a good approximation in various contexts and over a broad range of experimental settings. This model aims to give the slow evolution of the envelope of one monochromatic wave due to nonlinearity. Here, we naturally consider the same problem for the envelopes of two weakly resonant monochromatic waves. As for the nonlinear Schrödinger equation, which is integrable, we adopt an integrable model to describe the interaction of two waves. This is the system of two coupled nonlinear Schrödinger equations (Manakov model) with self- and cross-interactions that may be both defocussing and focussing. We first discuss the linear stability properties of the background by computing the spectrum for all values of the parameters such as coupling constants and amplitudes. In particular, we relate the instability bands to properties of the spectrum and compute the gain function (or growth rate). We also relate to the stability spectrum the value of the spectral variable, which corresponds to a rogue wave solution. In contrast with the nonlinear Schrödinger equation, different types of single rogue wave exist that correspond to different values of the spectral variable even in the same spectrum. For these critical values, which are completely classified, we give the corresponding explicit expression of the rogue wave solution that follows from the well known Darboux–Dressing transformation method. Although not all systems of two coupled nonlinear Schrödinger equations that have been derived in water wave dynamics are integrable, our investigation contributes to the understanding of new effects due to wave coupling, at least for model equations that, even if not integrable, are close enough to the model considered here. For instance, our findings lead to investigate rogue waves generated by instabilities due to self- and cross-interactions of defocusing type. An illustrative selection of two coupled rogue waves solutions is displayed