Lax System Research Articles

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

AbstractWe study the standing periodic waves in the semidiscrete integrable system modeled by the Ablowitz–Ladik (AL) equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a nonstandard linear Lax system, which is different from the standard Lax representation of the AL equation.

Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation

<abstract><p>The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $ N $-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation.</p></abstract>

Linear Instability of Breathers for the Focusing Nonlinear Schrödinger Equation

Relying upon tools from the theory of integrable systems, we discuss the linear instability of the Kuznetsov–Ma breathers and the Akhmediev breathers of the focusing nonlinear Schrödinger equation. We use the Darboux transformation to construct simultaneously the breathers and the exact solutions of the Lax system associated with the breathers. 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. While we do not attempt to prove completeness of eigenfunctions, we aim to determine the entire set of solutions of the linearized NLS equations generated by the Lax system in appropriate function spaces.

Hirota-Miwa equations on a time space scale

We derive a general time scale version of the Hirota–Miwa equation as a compatibility condition arising from a new version of a three-dimensional Lax system, construct its 3-soliton solutions, and consider various examples. We prove the invariance of the compatibility condition of the two-dimensional Lax system under a gauge transformation. The 3-soliton solution obtained here may be used in modeling in view of its variable graininess and a large number of free parameters.

Lax systems and nonlinear equations on a time-space scale

From the Lax system on a time-space scale, we derive Sine-Gordon, Burgers, and some other nonlinear equations. From the simplified two- and three-dimensional Lax systems on a time-space scale with variable nonvanishing graininess, we derive Korteweg–de Vries, Boussinesq, Krichever–Novikov, Hirota–Miwa, and other nonlinear equations on a time-space scale with soliton solutions. We also construct multi-soliton solutions for some equations by the direct method using the exponential functions on a time-space scale.

Lavado de activos y corrupción: análisis desde la perspectiva preventiva como medio para fortalecer la democracia

Este trabajo tiene como principal finalidad el análisis de los sistemas para abordar el problema social que genera en los sistemas democráticos el delito de lavado de activos y la corrupción, ya que debilitan las estructuras del Estado. Este estudio inicia con el análisis de doctrina internacional y estudios académicos realizados, tomando en consideración la disyuntiva en adoptar el sistema represivo o preventivo, analizándolos individualmente para identificar cuál sería el más conveniente para fortalecer la democracia en las naciones, considerando que la relación existente entre las medidas de orden preventivo y represivo es inmanente y este flagelo no concluye con la mera represión punitiva, incrementando las penas y sanciones, pero tampoco con un sistema permisivo o laxo. Llegando a determinar como una opción válida la adopción del sistema preventivo en torno a un delito que cada vez se innova con nuevas estratagemas, por lo que se requiere estudios constantes debido a esa especificidad de características particulares que requieren ser analizadas permanentemente.

Reforming Ethiopia’s Expropriation Law

Ethiopia is increasingly using expropriation as the single most important device to take land particularly from small landholders to supply it to corporate farmers and industrialists with a declared intention of boosting economic growth. This is happening in the context where expropriation laws are inadequate to protect peasants and pastoralists. The state is not paying cash compensation for land use rights, compensation for property on the land is paltry, and uniform rehabilitative schemes are absent. There are also no sufficient administrative and judicial mechanisms in place to restrain the government in exercising its power of expropriation. This is lax expropriation system that runs counter with the country`s Constitution which pledges tenure security for small landholders. This article recommends enhanced judicial scrutiny of expropriation, recasting land use rights as human rights, and emphasis on the quality of projects that necessitate expropriation. Ethiopia should expedite these and related measures in the interest of justice, securing the livelihoods of the masses and national stability. Key terms: Expropriation, public purpose, compensation, land use rights and rehabilitation

Nonintersecting Brownian motions on the unit circle

We consider an ensemble of $n$ nonintersecting Brownian particles on the unit circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at the same point and to return to that point after time $T$, but otherwise not to intersect. There is a critical value of $T$ which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as $n\to\infty$, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlev\'{e} II equation of size 2 $\times$ 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.

Bäcklund transformations, Lax system, conservation laws and multisoliton solutions for Jimbo–Miwa equation with Bell-polynomials

Based on binary Bell polynomial approach, the bilinear equation and Ba¨cklund transformations for (3+1)-dimensional Jimbo–Miwa equation are obtained. By virtue of Cole–Hopf transformation, Lax system is constructed by direct linearization of coupled system of binary Bell polynomials. Furthermore, infinite conservation laws are obtained from two field condition in quick and natural way. Finally, a test function of extended three wave method is used to construct multisoliton solutions via bilinear equation.

Two Lax systems for the Painlevé II Equation, and Two Related Kernels in Random Matrix Theory

We consider two Lax systems for the homogeneous Painlevé II equation: one of size $2\times 2$ studied by Flaschka and Newell in the early 1980s, and one of size $4\times 4$ introduced by Delvaux, Kuijlaars, and Zhang and Duits and Geudens in the early 2010s. We prove that solutions to the $4\times 4$ system can be derived from those to the $2\times 2$ system via an integral transform, and consequently relate the Stokes multipliers for the two systems. As corollaries we are able to express two kernels for determinantal processes as contour integrals involving the Flaschka--Newell Lax system: the tacnode kernel arising in models of nonintersecting paths and a critical kernel arising in a two-matrix model.

Connections defining representations of zero curvature and the solitons of sine-Gordon and Korteweg-de Vries equations

It is claimed that solutions of travelling-wave type (and, in particular, soliton solutions) of partial differential equations can be created by using connections defining representations of zero curvature. In this paper, we construct solitons of the sine-Gordon and Korteweg-de Vries equations. By previous results of the author, the connections defining representations of zero curvature for a given differential equation generate Backlund transformations for this equation. It can be shown that the well-known Lax system (the so-called Lax pair) for the Korteweg-de Vries equation is a special case of a Backlund system (i.e., the system of partial differential equations defining a Backlund transformation). Note that the creation of solitons by means of the inverse scattering method is in fact a creation of solitons by means of the Lax system (without using connections defining the representations of zero curvature from the very beginning). Moreover, the inverse scattering method is essentially more labor-consuming than the method suggested in the present paper. Further, it is not required to involve any physical notions when using the suggested method. In the final section of the paper, we consider the so-called 2-soliton solutions of sine-Gordon and Korteweg — de Vries equations. Here we systematically use the invariant analytic method developed by G. F. Laptev, which is well-known in differential geometry under the title of Cartan-Laptev method.

The relationship between the hyperbolic Nizhnik–Novikov–Veselov equation and the stationary Davey–Stewartson II equation

A Lax system in three variables is presented, two equations of which form the Lax pair of the stationary Davey–Stewartson II equation. With certain nonlinear constraints, the full integrability condition of this Lax system contains the hyperbolic Nizhnik–Novikov–Veselov equation and its standard Lax pair. The Darboux transformation for the Davey–Stewartson II equation is used to solve the hyperbolic Nizhnik–Novikov–Veselov equation, and global multi-soliton solutions are obtained. It is proved that the n × n soliton solution derived by the Darboux transformation of order n approaches zero uniformly and exponentially at spatial infinity and is asymptotic to n2 solitons at temporal infinity.

A Stiglerian View on Banking Supervision

The system of banking supervision in Europe is undergoing substantial reforms. According to Stigler's capture theory regulation often follows the preferences of producers. Therefore, the interests of the financial industry might be a major driving force for the ongoing supervisory reform debate. This paper identifies possible interests of the regulated industries: Either they might favour strict supervision to create barriers for entry and thus to reduce competitive pressure in their market. Or they might use their political influence to press for a lax and low-cost supervisory system. A cross-country data base on supervisory systems and financial structure allows the application of a three-step testing procedure. It turns out that the private interest view on regulation is indeed relevant and that the data is more compatible with a preference for laxity than with a barriers to entry view.

A Stiglerian View on Banking Supervision

The system of banking supervision in Europe is undergoing substantial reforms. According to Stigler's capture theory regulation often follows the preferences of producers. Therefore, the interests of the financial industry might be a major driving force for the ongoing supervisory reform debate. This paper identifies possible interests of the regulated industries: Either they might favour strict supervision to create barriers for entry and thus to reduce competitive pressure in their market. Or they might use their political influence to press for a lax and low-cost supervisory system. A cross-country data base on supervisory systems and financial structure allows the application of a three-step testing procedure. It turns out that the private interest view on regulation is indeed relevant and that the data is more compatible with a preference for laxity than with a barriers to entry view.

Finite Dimensional Integrable Hamiltonian Systems Associated with DSI Equation by Bargmann Constraints

The Davey-Stewartson I equation is a typical integrable equation in 2+1 dimensions. Its Lax system being essentially in 1+1 dimensional form has been found through nonlinearization from 2+1 dimensions to 1+1 dimensions. In the present paper, this essentially 1+1 dimensional Lax system is further nonlinearized into 1+0 dimensional Hamiltonian systems by taking the Bargmann constraints. It is shown that the resulting 1+0 dimensional Hamiltonian systems are completely integrable in Liouville sense by finding a full set of integrals of motion and proving their functional independence.

On the Restricted Toda and c-KdV Flows of Neumann Type**The project supported by the Chinese National Basic Research Project "Nonlinear Science" and the Doctoral Programme Foundation of Institution of High Education of China. The first author also thanks the National Natural Science Foundation of China (19801031) and "Qinglan Project" of Jiangsu Province of China; and the second author also thanks

It is proven that on a symplectic submanifold the restricted c-KdV flow is just the interpolating Hamiltonian flow of invariant for the restricted Toda flow, which is an integrable symplectic map of Neumann type. They share the common Lax matrix, dynamical r-matrix and system of involutive conserved integrals. Furthermore, the procedure of separation of variables is considered for the restricted c-KdV flow of Neumann type.

Numerical procedures based on Runge-Kutta methods for solving isospectral flows

This paper deals with the numerical solution of the Lax system L′ = [ B( L), L], L(0) = L 0 (∗), where L 0 is a constant symmetric matrix, B(·) maps symmetric matrices into skew-symmetric matrices, and [ B( L), L] is the commutator of B( L) and L. Here two different procedures, based on the approach recently proposed by Calvo, Iserles and Zanna (the MGLRK methods), are suggested. Such an approach is a computational form for the Flaschka formulation of (∗). Our numerical procedures consist in solving (∗) by a Runge-Kutta method, then, a single step of a Gauss-Legendre Runge-Kutta (GLRK) method may be applied to the Flaschka formulation of (∗). In the first procedure we compute the approximation of the Lax system by a continuous explicit RK method, instead, the second procedure computes the approximation of the Lax system by a GLRK method (the same method used for the Flaschka system). The computational costs have been derived and compared with the ones of the MGLRK methods. Finally, several numerical tests and computational comparisons will be shown.

Japan bans shingles drug after deaths

Nippon Shoji, a Japanese pharmaceutical company that marketed a drug for shingles but hid the fact that three people had died in clinical trials, has been shut down for three months by the government in Tokyo. Fifteen patients are alleged to have died within weeks of taking the drug when it first went on sale last autumn. The case has focused attention on Japan's lax system of testing drugs and on a medical profession that has been accused of sacrificing safety in the pursuit of profit. No criminal charges have been brought against Nippon Shoji's executives, and the drug, sorivudine, will still be available to those hospitals equipped to deal with its side effects. Sorivudine, which was marketed under the name Usevir, …

Lax representation of Riemann ellipsoids

The fluid dynamical equations of motion for rotating Riemann ellipsoids are presented as a Lax equation, X = [ F, X]. Riemann ellipsoids model rotating galaxies, stars, and atomic nuclei. The elements of the Lax pair X- F are matrices in the symplectic Lie algebra sp(3, R). The matrix X is formed from the inertia and virial momentum tensors; F is composed from the potential energy, angular velocity, and pressure tensors. The Kelvin circulation is conserved by the Lax system. Riemann ellipsoidal solutions are given by symplectic isospectral deformations.

A new hierarchy of nonlinear evolution equations and corresponding finite-dimensional completely integrable systems

A new isospectral problem and the corresponding hierarchy of nonlinear evolution equations are proposed. Under a constraint between the potentials and the eigenfunctions, the Lax system associated with the hierarchy of evolution equations is nonlinearized to be a hierarchy of finite-dimensional completely integrable systems in the Liouville sense.