Liouville Integrability Research Articles

A Liouville integrable hierarchy with four potentials and its bi-Hamiltonian structure

"We aim to construct a Liouville integrable Hamiltonian hierarchy from a specific matrix spectral problem with four potentials through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is exhibited by a bi-Hamiltonian structure explored by using the trace identity. Illustrative examples of novel four-component coupled Liouville integrable nonlinear Schr¨odinger equations and modified Korteweg-de Vries equations are presented."

Certain generalized Riemann–Liouville fractional integrals inequalities based on exponentially (h,m)-preinvexity

In this article, we define a new preinvex mapping known as the exponentially (h,m)-preinvex. We explore the properties of this mapping, and derive several Φk-Riemann–Liouville fractional integrals inequalities through exponentially (h,m)-preinvexity. Using these results, we present certain estimates for Hermite–Hadamard-type inequalities, and also deduce several inequalities for the fractional integrals of k-Riemann–Liouville, which are special cases of the main results. During this period, we give several numerical examples to further demonstrate the correctness of our main results. As applications, we propose a few inequalities according to special means.

Nearly integrable flows and chaotic tangles in the Dimits shift regime of plasma edge turbulence

Transitionally turbulent flows frequently exhibit spatiotemporal intermittency, reflecting a complex interplay between driving forces, dissipation, and transport present in these systems. When this intermittency manifests as observable structures and patterns in the flow, the characterization of turbulence in these systems becomes challenging due to the nontrivial correlations introduced into the statistics of the turbulence by these structures. In this work, we use tools from dynamical systems theory to study intermittency in the Dimits shift regime of the flux-balanced Hasegawa–Wakatani (BHW) equations, which models a transitional regime of resistive drift-wave turbulence relevant to magnetically confined fusion plasmas. First, we show in direct numerical simulations that turbulence in this regime is dominated by strong zonal flows and coherent drift-wave vortex structures, which maintain a strong linear character despite their large amplitude. Using the framework of generalized Liouville integrability, we develop a theory of integrable Lagrangian flows in generic fluid and plasma systems and discuss how the observed zonal flows plus drift waves in the BHW system exhibit a form of “near-integrability” originating from a fluid element relabeling symmetry. We further demonstrate that the BHW flows transition from integrability to chaos via the formation of chaotic tangles in the aperiodic Lagrangian flow, and establish a direct link between the “lobes” associated with these tangles and intermittency in the observed turbulent dissipation. This illustrates how utilizing tools from deterministic dynamical systems theory to study convective nonlinearities can explain aspects of the intermittent spatiotemporal structure exhibited by the statistics of turbulent fields.

Approximating fractional calculus operators with general analytic kernel by Stancu variant of modified Bernstein–Kantorovich operators

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative.

A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with generalized memory kernel in the presence of time delay

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α+1 where 0<α<1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis.

Four-component integrable hierarchies and their Hamiltonian structures

We aim to construct four-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation. The Liouville integrability of the resulting hierarchies are guaranteed through establishing Hamiltonian structures by the trace identity. Illustrative examples include novel four-component nonlinear Schrödinger type equations and modified Korteweg–de Vries type equations.

New Fractional Inequalities through Convex Functions and Comprehensive Riemann–Liouville Integrals

In most fields of applied sciences, inequalities are important in constructing mathematical systems and associated solution functions. Convexity also has a significant impact on an assortment of mathematical topics. By utilizing a comprehensive version of Riemann–Liouville integrals and the functions’ convexity condition, we present and prove novel fractional inequalities. According to the current literature, this work is a novel addition to the literature, and the proposed technique for addressing fractional inequalities issues is straightforward and simple to execute. It is also easy to see that all of the inequalities that have been developed are inclusive and may be reduced to a variety of other inequalities that have been proposed in the literature. Additionally, certain numeric examples with graphs are provided to support the theoretical results.

Semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries equations

The ubiquitous Korteweg–de Vries equation is widely used in oceanic, atmospheric dynamics, and fluid mechanics and can explain almost all kinds of physical phenomena of waves, while explicit solutions are studied in nonlinear mechanics and optics. We investigate explicit solutions for the semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries (cmKdV) equations, which describe sound waves in an inharmonic lattice. A semi-discrete four-component Frobenius-coupled cmKdV equation is presented by utilizing a zero-curvature equation. We use two different reduction methods to get a semi-discrete local Frobenius-coupled cmKdV equation and a semi-discrete nonlocal Frobenius-coupled cmKdV equation. In the paper, we study the above two new types of semi-discrete Frobenius-coupled cmKdV equations. Then the bi-Hamiltonian structures of the above two types of equations are constructed, which show that the two types of equations possess Liouville integrability. The covariant properties of the both new equations are shown by constructing their corresponding Darboux transformation (DT), respectively. Soliton solutions, semi-rational soliton solutions, and rogue wave solutions of the above two types of equations are constructed by N-fold DT and generalized (n,N−n)-fold DT. Three-dimensional plots and density profiles of these explicit solutions are presented. The characteristics of these figures demonstrate the relationship between the above two types of equations.

A six-component integrable hierarchy and its Hamiltonian formulation

The aim of this paper is to construct a six-component integrable hierarchy associated with a matrix spatial spectral problem of arbitrary order. The adopted method is the zero curvature formulation. The corresponding Hamiltonian formulation is furnished by using the trace identity, which guarantees the Liouville integrability for the resulting hierarchy. Two illustrative examples of integrable equations of lower orders are six-component coupled nonlinear Schrödinger equations and modified Korteweg–de Vries equations.

Local Yang–Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang–Baxter equation. Sergeev (1998 Lett. Math. Phys. 45 113–9) presented classification results on three-dimensional tetrahedron maps obtained from the local Yang–Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang–Baxter equation for some simple matrix-functions, we show that there are (non-unique) solutions to the local Yang–Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang–Baxter equation, we obtain several new birational tetrahedron maps with Lax representations and invariants, including maps on arbitrary groups, a nine-dimensional map associated with a Darboux transformation for the derivative nonlinear Schrödinger equation, and a nine-dimensional generalisation of the three-dimensional Hirota map.

New Versions of Midpoint Inequalities Based on Extended Riemann–Liouville Fractional Integrals

This study aims to prove some midpoint-type inequalities for fractional extended Riemann–Liouville integrals. Crucial equality is proven to build new results. Using this equality, several midpoint-type inequalities are established via differentiable convex functions and the proposed extended fractional operators. To be more specific, the well-known Hölder, Jensen, and power mean integral inequalities are employed in the demonstrated inequalities. Additionally, many remarks based on specific selections of the main results are presented. Moreover, to illustrate the key conclusions, a few instances are provided.

Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem

This paper is concerned with a two-dimensional fractional optimal control problem whose governing equations are distributed order fractional differential equations in the Caputo sense. A generalized fractional-order Legendre wavelet method has been used to solve the two-dimensional distributed-order fractional optimal control problem. An exact formula for the Riemann–Liouville integration of generalized fractional-order Legendre wavelet has been derived by using regularized beta functions. This formula and the two-dimensional Gauss–Legendre integration formula have been used to solve the two-dimensional distributed order fractional optimal control problem. Moreover, an L2-error estimate in the approximation of an unknown function with a generalized fractional-order Legendre wavelet has been derived and the estimated order has been verified for a given function. Furthermore, convergence analysis for the proposed method has been presented. In the last, two test problems have been considered to illustrate the efficiency of the proposed method.

Higher-Order Matrix Spectral Problems and Their Integrable Hamiltonian Hierarchies

Starting from a kind of higher-order matrix spectral problems, we generate integrable Hamiltonian hierarchies through the zero-curvature formulation. To guarantee the Liouville integrability of the obtained hierarchies, the trace identity is used to establish their Hamiltonian structures. Illuminating examples of coupled nonlinear Schrödinger equations and coupled modified Korteweg–de Vries equations are worked out.

Geodesic motion on the symplectic leaf of SO(3) with distorted e(3) algebra and Liouville integrability of a free rigid body

The solutions to the Euler–Poisson equations are geodesic lines of SO(3) manifold with the metric determined by inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We calculate its explicit form and confirm that it differs from the algebra e(3). The obtained Poisson brackets are used to demonstrate the Liouville integrability of a free rigid body. The general solution to the Euler–Poisson equations is written in terms of exponential of the Hamiltonian vector field.

The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

On the integrability of the vocal fold model

In this work the algebraic characterization of the Darboux, Liouville, and Puiseux first integrals of the vocal fold model is given. The multiple Puiseux solutions, also defined in Puiseux integrability, are studied for the first time.

Analysis of the multi-phenomenal nonlinear system : Testing Integrability and detecting chaos

As a large extension in Hamiltonian form, the system of a PT symmetric dimer of coupled nonlinear oscillators is developed. This system provides an explanation for a number of problems with Hamiltonian dynamics. Integrability is evaluated in the Painlevé sense of the system. The system reported twelve P-cases. First integrals of planar motion are constructed explicitly for each integrable case to show the Liouvillian integrability of the equations of motion. A mixture of numerical approaches is used to test the theoretical conclusions in order to identify the nature of orbits and evaluate the system’s transition from order to chaos. These techniques consist of the Poincaré Section Surface, the maximum Lyapunov Exponent, and the Smaller Alignment Index.

Lotka–Volterra lattice system: N-fold Darboux transformation, the corresponding integrable lattice family and bi-Hamiltonian structure

An one-fold Darboux transformation for the Lotka–Volterra lattice system is first established using a proper gauge transformation matrix. Then, as a result of the N times one-fold Darboux transformation, the corresponding N-fold Darboux transformation of the Lotka–Volterra lattice system is presented, and two exact solution are obtained by the resulting Darboux transformation. Hereafter, its the corresponding iso-spectral integrable lattice family is derived. Using the trace identity, bi-Hamiltonian structure of the Lotka–Volterra integrable family is established, and its Liouville integrability is proven.

Bifurcation Diagram of the Model of a Lagrange Top with a Vibrating Suspension Point

The article considers a model system that describes a dynamically symmetric rigid body in the Lagrange case with a suspension point that performs high-frequency oscillations. This system, reduced to axes rigidly connected to the body, after the averaging procedure, has the form of the Hamilton equations with two degrees of freedom and has the Liouville integrability property of a Hamiltonian system with two degrees of freedom, which describes the dynamics of a Lagrange top with an oscillating suspension point. The paper presents a bifurcation diagram of the moment mapping. Using the bifurcation diagram, we presented in geometric form the results of the study of the problem of stability of singular points, in particular, singular points of rank zero and rank one.

Synchronization of fractional-order chaotic networks in Presnov form via homogeneous controllers

The Presnov decomposition for continuously differential vector fields consists of a gradient of a scalar field (dissipative term) and a vector field orthogonal to a radial vector of the origin (circulative or non-dissipative term). This paper uses the Presnov decomposition associated with a fractional-order Lorenz family to design algorithms that solve the multi-synchronization problem. Assuming that the whole state is known, two active control solutions are proposed to achieve the desired synchronization. Both control algorithms take advantage of homogeneous functions to have a powerful and simple Lyapunov analysis to show Mittag-Leffler stabilization in the first algorithm. In contrast, the second one achieves synchronization before a predefined time defined by the user. The combination of homogeneous functions and Riemann–Liouville integral in the controllers improve the behavior of the dynamic systems due to the impact on the robustness properties in the control scheme. Numerical simulations are presented to show the reliability of the proposed methods.