Liouville Integrability Research Articles

Fractional Singular Differential Systems of Lane–Emden Type: Existence and Uniqueness of Solutions

A coupled system of singular fractional differential equations involving Riemann–Liouville integral and Caputo derivative is considered in this paper. The question of existence and uniqueness of solutions is studied using Banach contraction principle. Furthermore, the question of existence of at least one solution is discussed. At the end, an illustrative example is given in details.

Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

Classical integrability of the zigzag model

The zigzag model is a relativistic $N$-body system arising in the high energy limit of the worldsheet scattering in adjoint two-dimensional QCD. We prove classical Liouville integrability of this model by providing an explicit construction of $N$ charges in involution. Furthermore, we also prove that the system is maximally superintegrable by constructing $N-1$ additional independent charges. All of these charges are piecewise linear functions of coordinates and momenta. The classical time delays are determined algebraically from this integrable structure. The resulting $S$-matrix is the same as in the $N$-particle subsector of a massless $T\bar{T}$ deformed fermion.

Poisson Quasi-Nijenhuis Manifolds and the Toda System

The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by an example with $3$ degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we prove that the closed (or periodic) $n$-particle Toda lattice can be framed in such a geometrical structure, and its well-known integrals of the motion can be obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that is, a tensor field $N$ of type $(1,1)$ defined on the phase space of the lattice. This example and some of its generalizations are used to understand whether one can define in a reasonable sense a notion of {\em involutive\} Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non periodic) and the closed Toda systems is also framed in the context of a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds.

Integrable asymmetric AKNS model with multi-component

A novel integrable asymmetric coupling of the Ablowitz-Kaup-Newell-Segur (AKNS) system was generated by the completion process of the integrable coupling. We proved the Liouville integrability of the AKNS integrable coupling by deriving its bi-Hamiltonian structures applying the variational identity. Nextly, we investigated the generalized symmetry and infinitely many conservation laws by Hereman method for the integrable coupling system considered in this paper. Finally, we presented the explicit solutions of the resulting integrable coupling system by constructing Darboux transformation for the first time.

Liouvillian integrability of the three-dimensional generalized Hénon–Heiles Hamiltonian

In this paper, we report about three cases of integrability in sense of Liouville for three-dimensional generalized Henon–Heiles Hamiltonian. This also allow to get explicitly integrals of motions for each case. On the other hand, this paper investigates the phase space structure numerically with Poincare surfaces of section and 3D projections which allow to verify that the analytical results are in agreement with the computations.

Linearizability conditions for the Rayleigh-like oscillators

In this work we consider a family of nonlinear oscillators that is cubic with respect to the first derivative. Particular members of this family of equations often appear in numerous applications. We solve the linearization problem for this family of equations, where as equivalence transformations we use generalized nonlocal transformations. We explicitly find correlations on the coefficients of the considered family of equations that give the necessary and sufficient conditions for linearizability. We also demonstrate that each linearizable equation from the considered family admits an autonomous Liouvillian first integral, that is Liouvillian integrable. Furthermore, we demonstrate that linearizable equations from the considered family does not possess limit cycles. Finally, we illustrate our results by two new examples of the Liouvillian integrable nonlinear oscillators, namely by the Rayleigh–Duffing oscillator and the generalized Duffing–Van der Pol oscillator.

Particular superintegrability of 3-body (modified) Newtonian gravity

It is found explicitly 5 Liouville integrals in addition to total angular momentum which Poisson commute with Hamiltonian of 3-body Newtonian Gravity in [Formula: see text] along the remarkable figure-8-shape trajectory discovered by Moore-Chenciner-Montgomery. It is verified that they become constants of motion along this trajectory. Hence, 3-body choreographic motion on figure-8-shape trajectory in [Formula: see text] Newtonian gravity (Moore, 1993), as well as in [Formula: see text] modified Newtonian gravity by Fujiwara et al., is maximally superintegrable. It is conjectured that any 3-body potential theory that admits Figure-8-shape choreographic motion is superintegrable along the trajectory.

Hyperbolic Riesz B-Potential and Solution of an Iterated Non-Homogeneous B-Hyperbolic Equation

Marcel Riesz has created a new method to solve non-homogeneous linear equations, generalizing the fractional Riemann–Liouville integral. We generalize and apply this method to solve linear equations with Bessel operators acting with respect to all variables. This method includes the overcoming of difficulties of the theory of differential equations, caused by the occurrence of divergent integrals. Namely, in some cases (for example, for hyperbolic equations), it is necessary to use the analytical continuation of a potential analytically depending on a parameter.

A superlinear numerical scheme for multi-term fractional nonlinear ordinary differential equations

In this paper, we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations (MTFNODEs). First, the presented problem is equivalently transformed into its integral form with multi-term Riemann–Liouville integrals. Second, the compound product trapezoidal rule is used to approximate the fractional integrals. Then, the unconditional stability and convergence with the order [Formula: see text] of the proposed scheme are strictly established, where [Formula: see text] and [Formula: see text] are the maximum and the second maximum fractional indexes in the considered MTFNODEs, respectively. Finally, two numerical examples are provided to support the theoretical results.

On a Fractional Operator Combining Proportional and Classical Differintegrals

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

Grüss-type integrals inequalities via generalized proportional fractional operators

In the article, we deal with the generalized proportional fractional integral, establish several kinds of inequalities such as Gruss-type and certain other inequalities by use of generalized proportional fractional integral. Moreover, several special cases are discussed. Also, we derive certain particular results by utilizing the connection between generalized proportional fractional integral and Riemann–Liouville integral. Furthermore, an illustrative example is presented to support our outcomes.

Bi-rational maps in four dimensions with two invariants

In this paper we present a class of four-dimensional bi-rational maps with two polynomial invariants of particular degree patterns. We discuss the integrability properties of these maps from the point of view of degree growth and Liouville integrability.

On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

AbstractWe present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

Fractional Sturm–Liouville eigenvalue problems, I

We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann–Liouville type. We then solve a Dirichlet type Sturm–Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann–Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann–Liouville integrals at those end-points. For each $$1/2<\alpha <1$$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $$\alpha \rightarrow 1^-$$, and that the fractional operator converges to an ordinary two term Sturm–Liouville operator as $$\alpha \rightarrow 1^-$$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $$\alpha $$.

Induced Dynamics

Construction of new integrable systems and methods of their investigation is one of the main directions of development of the modern mathematical physics. Here we present an approach based on the study of behavior of roots of functions of canonical variables with respect to a parameter of simultaneous shift of space variables. Dynamics of singularities of the KdV and Sinh—Gordon equations, as well as rational cases of the Calogero– Moser and Ruijsenaars–Schneider models are shown to provide examples of such induced dynamics. Some other examples are given to demonstrates highly nontrivial collisions of particles and Liouville integrability of induced dynamical systems.

Generalized Integral Inequalities for Hermite–Hadamard-Type Inequalities via s-Convexity on Fractal Sets

In this article, we establish new Hermite–Hadamard-type inequalities via Riemann–Liouville integrals of a function ψ taking its value in a fractal subset of R and possessing an appropriate generalized s-convexity property. It is shown that these fractal inequalities give rise to a generalized s-convexity property of ψ . We also prove certain inequalities involving Riemann–Liouville integrals of a function ψ provided that the absolute value of the first or second order derivative of ψ possesses an appropriate fractal s-convexity property.

Some tastings in Morales-Ramis theory

In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system.

Notes on “A integrable lattice hierarchy based on Suris system: N-fold Darboux transformation and conservation laws” [Nonlinear. Dyn. (2018) 91:625–639

We point out that the Hamiltonian structure of the integrable lattice hierarchy in “A integrable lattice hierarchy based on Suris system: N-fold Darboux transformation and conservation laws” [Nonlinear. Dyn. (2018) 91:625–639] is not correct. Then, we establish a correct Hamiltonian structure by the trace identity. Further, we also prove that the integrable lattice hierarchy has bi-Hamiltonian structure. Thus, the Liouville integrability of the integrable lattice hierarchy is obtained.

On the global dynamics and integrability of the Chemostat system

We study a Chemostat system of the form ẋ=−qx+R̃K+yxy,ẏ=(c̃−y)q−R̃ã(K+y)xy,where q>0, R̃>0, K>0, c̃>0 and ã≠0. This system appears in competition modelling in biology. We describe its global dynamics on the Poincaré disc and study its Liouvillian integrability. For the first topic we use the well-known Poincaré compactification theory and for the second one we make use of the Puiseux series to derive the structure of all the irreducible invariant algebraic curves.