arXiv subjects

Ghorbanali Haghighatdoost

Publications and source records attributed to Ghorbanali Haghighatdoost.

11 recordsLinked to original sources

arXiv2026

Lie symmetry analysis and similarity reductions for the tempered-fractional Keller Segel system

We perform a Lie symmetry analysis on the tempered-fractional Keller Segel (TFKS) system, a chemo-taxis model incorporating anomalous diffusion. A novel approach is used to handle the nonlocal nature of tempered fractional operators. By deriving the full set of Lie point symmetries and identifying the optimal one-dimensional subalgebras, we reduce the TFKS PDEs to ordinary differential equations (ODEs), yielding new exact solutions. These results offer insights into the long-term behavior and aggregation dynamics of the TFKS model and present a methodology applicable to other tempered fractional differential equations.

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arXiv2026

Similarity Solutions for the Flux limited Keller Segel System with Time Varying Chemical Decay Rate

We investigate a one dimensional flux limited Keller Segel system (FLKS) in which the chemical decay rate is allowed to vary explicitly in time, a feature motivated by enzymatic regulation and environmental variability in chemotactic signalling. Treating the decay rate as an arbitrary function, we carry out a systematic Lie symmetry analysis of the resulting PDE system and employ equivalence transformations to perform a complete group classification, we identify the kernel symmetry algebra admitted for arbitrary decay functions and determine three distinguished cases that extend the symmetry algebra constant decay rates, inverse time (power law) decay, and exponential decay. For each case, we construct an optimal system of subalgebras and derive the corresponding similarity reductions. Finally, we find some explicit solutions for our FLKS model. Our results provide a rigorous mathematical foundation for understanding which temporal decay patterns admit similarity reductions, thereby enabling analytical progress on flux limited chemotaxis models with realistic time varying degradation mechanisms.

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arXiv2026

Lie symmetry analysis of a 2+1-dimensional flux limited Keller Segel system

We investigate a two dimensional flux limited Keller Segel FLKS system using classical Lie symmetry analysis. The complete Lie point symmetry algebra is determined and shown to consist of temporal and spatial translations together with planar rotations. An optimal system of one-dimensional subalgebras is constructed, yielding stationary, translationally invariant, radially symmetric, travelling-wave, and rotatingwave reductions. The corresponding reduced equations are derived systematically. The results demonstrate that the flux limiting mechanism eliminates the scaling symmetries of the classical Keller Segel model and significantly restricts the class of admissible invariant solutions. Exact spatially homogeneous equilibrium solutions are determined, and integral representations for stationary radial and travelling-wave profiles are derived.

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arXiv2026

Euler Poincare Dynamics and Control on Lie Groupoids

We extend the Euler Poincare formalism from Lie groups to Lie groupoids for optimal control problems. While Lie algebroids provide the standard infinitesimal framework, the groupoid formulation enables global trajectory reconstruction and naturally accommodates systems with non-trivial base manifolds. We derive the reduced EP equations on a Lie groupoid, specialize to the trivial groupoid, and illustrate the framework with a generalized rigid body on the sphere. A biological application to collective cell migration on a spherical tissue shows that the coupled dynamics of spatial migration and internal polarity rearrangement leads to optimal migration times of approximately 74 minutes, consistent with experimental observations. This work demonstrates how Lie groupoid methods broaden geometric control theory beyond standard rigid body systems.

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arXiv2026

Right invariant Poisson Nijenhuis structures on Lie groupoids Correspondence and Classification

In this paper, we introduce right-invariant Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called (Poisson bivector, Nijenhuis operator) structures. Also, we present a one-to-one correspondence between (Poisson bivector, Nijenhuis operator) structures on Lie algebroids with (Poisson, Nijenhuis) structures on their Lie groupoids under certian conditions. Also, we give some illustrative examples .

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arXiv2026

Poisson Hamiltonian Pontryagin Dynamics and Optimal Control of Mechanical Systems on Lie Groupoids

We develop a Poisson Hamiltonian formulation of Pontryagin dynamics for optimal control of mechanical systems on Lie groupoids. The reduced dynamics is formulated intrinsically on the dual Lie algebroid endowed with its canonical linear Poisson structure and evolves on its symplectic leaves. The main result of this work shows that symplectic leaves, rather than coadjoint orbits, provide the natural reduced phase spaces for Pontryagin dynamics on Lie groupoids. Under suitable regularity assumptions, we prove the equivalence between the variational formulation of the optimal control problem and the associated Poisson Hamiltonian Pontryagin system, and we show that groupoid invariant Lagrangians lead to reduced optimality conditions of Euler Poincare type. Several mechanical examples, including systems with configuration dependent inertia and local symmetries, are presented to illustrate the theory.

math.OC↗

arXiv2025

Geometric Dynamical Systems

This article provides a conceptual and historical review of the evolution of integrable Hamiltonian systems from the Moscow School of A. T. Fomenko to the emerging Azarbaijan School of Geometric Dynamical Systems founded by the author. Beginning with the topological classification of integrable systems through Liouville foliations, atoms, and molecular invariants, the paper traces how these geometric ideas evolved into modern frameworks based on Lie groupoids, Lie algebroids, and fractional calculus. The author s doctoral dissertation at Moscow State University extended 2004 the Fomenko theory to new integrable systems on so(4), constructing novel topological molecules and describing the hierarchy of singularities and bifurcations. Upon his return to Iran, he established a comprehensive research program at Azarbaijan Shahid Madani University, integrating topology, geometry, and analysis to form a coherent Iranian branch of the global theory of integrable systems. This program unifies the classical and the modern: from the Euler Poincare and Hamilton Jacobi formalisms on Lie groupoids and algebroids, to fractional and quantum mechanical models involving Hopf and C algebras. The paper emphasizes conceptual synthesis over computation, showing how integrable geometry has transformed from a purely mechanical theory into a universal language connecting topology, control theory, and quantum structures.

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arXiv2024

A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoids

As we said in our previous work [4], the main idea of our research is to introduce a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called coadjoint Lie groupoids. In this paper, we will examine the relationship between structural mappings of the Lie algebroid associated to Lie groupoid and co-adjoint Lie algebroid. Also, we try to construct and define integrabiliy of Hamiltonian system on the co-adjoint Lie groupoids. In addition, we show that co-adjoint Lie groupoid associated to a symplectic Lie groupoid is a symplectic Lie groupoid.

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arXiv2024

Optimal control problems on the co-adjoint Lie groupoids

In this work we study the invariant optimal control problem on Lie groupoids. We show that any invariant optimal control problem on a Lie groupoid reduces to its co-adjoint Lie algebroid. In the final section of the paper, we present an illustrative example.

math.OC↗

arXiv2018

Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups

We study {\em right-invariant (resp., left-invariant) Poisson quasi-Nijenhuis structures} on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called {\em r-qn structures} on the corresponding Lie algebra $\mathfrak g$. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all $r$-$qn$ structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between $r$-$qn$ structures and the generalized complex structures on the Lie algebras $\mathfrak g$ and also the solutions of modified Yang-Baxter equation on the double of Lie bialgebra $\mathfrak g\oplus\mathfrak g^*$. The results are applied to some relevant examples.

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arXiv2018

Invariant Poisson-Nijenhuis structures on Lie groups and classification

We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that $r$-$n$ structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two compatible r-matrices from which one is invertible determine an $r$-$n$ structure. We classify, up to a natural equivalence, all $r$-matrices and all $r$-$n$ structures with invertible $r$ on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by $r$-matrices on a phase space whose symmetry group is Lie group $G$, can be specifically determined.

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