Inverse Spectral Method Research Articles

Integration of a sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions

In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.

INTEGRATION OF A NONLINEAR HIROTA TYPE EQUATION WITH ADDITIONAL TERMS

In this paper, the inverse spectral problem method is used to integrate a nonlinear Hirota-type equation with additional terms in the class of periodic functions. The evolution of the spectral data of the periodic Dirac operator is introduced, the coefficient of Dirac operator is a solution to the nonlinear Hirota type equation with additional terms. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-zone functions is proved.

Интегрирование нелинейного уравнения Шредингера отрицательного порядка с нагруженным членом в классе периодических функций

In this paper, we consider the loaded negative order nonlinear Schrodinger equation (NSE) in the class of periodic functions. It is shown that the loaded negative order nonlinear Schrodinger equation can be integrated by the inverse spectral problem method. The evolution of the spectral data of the Dirac operator with a periodic potential associated with the solution of the loaded negative order nonlinear Schrodinger equation is determined. The results obtained make it possible to apply the inverse problem method to solve the loaded negative order nonlinear Schrodinger equation in the class of periodic ones. Important corollaries are obtained about the analyticity and period of the solution concerning the spatial variable.

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and using the first trace formula satisfies the Liouville equation.

Zadacha Koshi dlya nagruzhennogo uravneniya Kortevega–de Friza v klasse periodicheskikh funktsiy

The inverse spectral problem method is applied to finding a solution of the Cauchyproblem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded termsand a derivation of an analog of Dubrovin’s system of differential equations are proposed. Itis shown that the sum of a uniformly convergent function series constructed by solving theDubrovin system of equations and the first trace formula actually satisfies the loaded nonlinearKorteweg–de Vries equation. In addition, we prove that if the initial function is a real π-periodicanalytic function, then the solution of the Cauchy problem is a real analytic function in thevariable x as well, and also that if the number π/n, n ∈ N, n ≥ 2, is the period of the initialfunction, then the number π/n is the period for solving the Cauchy problem with respect to thevariable x.

Periodic solutions of the modified Korteweg–de Vries equation in hemodynamics

In this paper, we consider the periodic solution of the modified Korteweg-de Vries equation used for studies of hemodynamic processes. It is shown that the modified Korteweg-de Vries equation can be integrated by the inverse spectral problem method. The evolution of the spectral data of the Dirac operator with a periodic potential associated with the solution of the modified Korteweg-de Vries equation is determined. The obtained results substantiate the applicability of the inverse problem method for solving the modified Korteweg-de Vries equation for studying the laws of hemodynamics.

Analytical simulations of the Fokas system; extension (2 + 1)-dimensional nonlinear Schrödinger equation

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

Integration Of A Higher-Order Nonlinear SchröDinger System with a Self-Consistent Source in the Class of Periodic Functions

We use the inverse spectral method to integrate a higher-order nonlinear Schrodinger system with a self-consistent source in the class of periodic functions.

Solving the Periodic Toda-Type Chain with a Self-Consistent Source

In this article, we explore the periodic Toda-type chain. The aim of this work is to obtain representations for the solutions of the periodic Toda-type chain with self-consistent source within the framework of the inverse spectral method for the discrete Hill equation. An efficient method for integrating the periodic Toda-type chain with self-consistent source is presented. The results can be used in modeling special types of electric transmission lines.

A series-form solution of the coupled nonlinear equations by the method of directly defined inverse mapping and SRM

We derive a series-form solution of the coupled nonlinear equations by the method of directly defined inverse mapping (MDDIM) and spectral relaxation method (SRM). Then, via the method of directly defined inverse mapping with the series-form solution firstly reported in this paper, we can obtain the theoretical and approximate analytical solution that is laid out to explore the features of boundary layer analysis of the nanofluid flow due to a bidirectional stretching sheet under magnetic field along with thermal radiation. Three water-based nanofluids with nanoparticles of Al2O3, TiO2 and CuO play an important role in the solutions. Based on the MDDIM and SRM, the convergence and accuracy of the proposed problems are illustrated, which exhibit meaningful structures. We has been proposed the convergence controlling relaxation parameter () with the help of which the convergence rate and accuracy of the MDDIM are highly improved.

Direct and inverse problems for vector logarithmic potentials with external fields

We consider extremal problems for the energy of the logarithmic potential with external fields closely related with the inverse spectral problem method. The method is based on the relations between the external field and the supports of the equilibrium measures which were discovered in the pioneering papers of Rakhmanov, Saff, Mhaskar and Buyarov (RSMB-method). We propose a generalization of the RSMB-method for the vector of measures with matrix of interaction between components.

Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations

We derive explicit formulas for the characteristic curves associated with the multipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov equations. Such a curve traces the path of a fluid particle whose instantaneous velocity equals the elevation of the wave at that point (or the square of the elevation, in the Novikov case). The peakons themselves follow characteristic curves, and the remaining characteristic curves can be viewed as paths of 'ghostpeakons' with zero amplitude; hence, they can be obtained as solutions of the ODEs governing the dynamics of multipeakon solutions. The previously known solution formulas for multipeakons only cover the case when all amplitudes are nonzero, since they are based upon inverse spectral methods unable to detect the ghostpeakons. We show how to overcome this problem by taking a suitable limit in terms of spectral data, in order to force a selected peakon amplitude to become zero. Moreover, we use direct integration to compute the characteristic curves for the solution of the Degasperis-Procesi equation where a shockpeakon forms at a peakon-antipeakon collision. In addition to the theoretical interest in knowing the characteristic curves, they are also useful for plotting multipeakon solutions, as we illustrate in several examples.

Algorithm for Solving the Cauchy Problem for One Infinite-Dimensional System of Nonlinear Differential Equations

The Cauchy problem for an infinite-dimensional system of nonlinear evolution equations, which is a generalization of the Langmuir chain, is considered. The global solvability of the problem in the class of rapidly decreasing functions is established. By the inverse spectral method, an algorithm for constructing the solution is obtained.

The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type

AbstractThe tri‐Hamiltonian splitting method applied to the Hirota hierarchy has yielded two U(1)‐invariant nonlinear Partial Differential Equations (PDEs) that admit peakons (nonsmooth solitons). In the present paper, these two peakon PDEs are generalized to a family of U(1)‐invariant peakon PDEs parameterized by the real projective line . All equations in this family are shown to possess conservative peakon solutions (whose Sobolev norm is time invariant). The Hamiltonian structure for the sector of conservative peakons is identified and the peakon Ordinary Differential Equations (ODEs) are shown to be Hamiltonian with respect to several Poisson structures. As the main result, it is shown that inverse spectral methods allow one to solve explicitly the dynamics of conservative peakons using explicit solutions to a certain interpolation problem. The graphs of multipeakon solutions confirm the existence of multipeakon breathers as well as asymptotic formation of pairs of two peakon bound states in the nonperiodic time domain.

A Potential-Based Inverse Spectral Method to Noninvasively Localize Discordant Distributions of Alternans on the Heart From the ECG.

Image the spatial distribution of TWA on the heart surface from ECG measurements. We introduced the inverse spectral method (ISM), a tailored inverse (or ElectroCardioGraphic Imaging) solution designed specifically to noninvasively image cases of TWA on the heart. We evaluate the ISM on its capacity to reliably detect the spatial distributions of TWA compared against a standard TWA detection method applied directly to the electrograms on the heart surface. We report on results from both a series of synthetic simulations of TWA generated using the ECGSIM software and a set of continuous epicardial surface voltage recordings from a canine experiment. ISM detected TWA distributions that matched the phase of the true underlying out-of-phase regions over and of the heart surface, respectively. Our results suggest that ISM is capable of reliably detecting the different regions present in a TWA distribution across a wide variety of TWA locations on the heart in simulation and in the face of transients and nonidealities in the canine recordings.

Solving Dirac equations on a 3D lattice with inverse Hamiltonian and spectral methods

A new method to solve the Dirac equation on a 3D lattice is proposed, in which the variational collapse problem is avoided by the inverse Hamiltonian method and the fermion doubling problem is avoided by performing spatial derivatives in momentum space with the help of the discrete Fourier transform, i.e., the spectral method. This method is demonstrated in solving the Dirac equation for a given spherical potential in 3D lattice space. In comparison with the results obtained by the shooting method, the differences in single particle energy are smaller than $10^{-4}$~MeV, and the densities are almost identical, which demonstrates the high accuracy of the present method. The results obtained by applying this method without any modification to solve the Dirac equations for an axial deformed, non-axial deformed, and octupole deformed potential are provided and discussed.

Erosion-driven uplift in the Gamburtsev Subglacial Mountains of East Antarctica

The relative roles of climate and tectonics in mountain building have been widely debated. Central to this debate is the process of flexural uplift in response to valley incision. Here we quantify this process in the Gamburtsev Subglacial Mountains, a paradoxical tectonic feature in cratonic East Antarctica. Previous studies indicate that rifting and strike-slip tectonics may have provided a key trigger for the initial uplift of the Gamburtsevs, but the contribution of more recent valley incision remains to be quantified. Inverse spectral (free-air admittance and Bouguer coherence) methods indicate that, unusually for continents, the coherence between free-air gravity anomalies and bedrock topography is high (>0.5) and that the elastic thickness of the lithosphere is anomalously low (<15 km), in contrast to previously reported values of up to ∼70 km. The isostatic effects of two different styles of erosion are quantified: dendritic fluvial incision overprinted by Alpine-style glacial erosion in the Gamburtsevs and outlet glacier-type selective linear erosion in the Lambert Rift, part of the East Antarctic Rift System. 3D flexural models indicate that valley incision has contributed ca. 500 m of peak uplift in the Gamburtsevs and up to 1.2 km in the Lambert Rift, which is consistent with the present-day elevation of Oligocene–Miocene glaciomarine sediments. Overall, we find that 17–25% of Gamburtsev peak uplift can be explained by erosional unloading. These relatively low values are typical of temperate mountain ranges, suggesting that most of the valley incision in the Gamburtsevs occurred prior to widespread glaciation at 34 Ma. The pre-incision topography of the Gamburtsevs lies at 2–2.5 km above sea-level, confirming that they were a key inception point for the development of the East Antarctic Ice Sheet. Tectonic and/or dynamic processes were therefore responsible for ca. 80% of the elevation of the modern Gamburtsev Subglacial Mountains.

On Isospectral Deformations of an Inhomogeneous String

In this paper we consider a class of isospectral deformations of the inhomogeneous string boundary value problem. The deformations considered are generalizations of the isospectral deformation that has arisen in connection with the Camassa-Holm equation for the shallow water waves. It is proved that these new isospectral deformations result in evolution equations on the mass density whose form depends on how the string is tied at the endpoints. Moreover, it is shown that the evolution equations in this class linearize on the spectral side and hence can be solved by the inverse spectral method. In particular, the problem involving a mass density given by a discrete finite measure and arbitrary boundary conditions is shown to be solvable by Stieltjes' continued fractions.

On the Solutions of Some Boundary Value Problems for the General Kdv Equation

This paper is concerned with a class of partial differential equations, which are the linear combinations, with constant coefficients, of the classical flows of the KdV hierarchy. A boundary value problem with inhomogeneous boundary conditions of a certain special form is studied. We construct some class of solutions of the problem using the inverse spectral method.

A class of complex solutions to the finite Toda lattice

In this paper, a class of complex-valued solutions to the finite Toda lattice is constructed by using the inverse spectral method. The corresponding Lax operator is a finite complex Jacobi matrix. As the initial values there are taken such complex numbers that the corresponding Jacobi matrix has a simple spectrum. Some examples are given.