Homotopy Operator Research Articles

Lepage Equivalents and the Variational Bicomplex

We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper we investigate the role of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators.

The Higher Structure of Unstable Homotopy Groups

Abstract We construct certain unstable higher order homotopy operations indexed by the simplex categories of $\boldsymbol{\Delta }^{n}$ for $n\geq 2$, and prove that all elements in the homotopy groups of a wedge of spheres are generated under such operations by Whitehead products and the group structure. This provides a stronger unstable analogue of Cohen’s theorem on the decomposition of stable homotopy.

Integrability Properties of the Slepyan–Palmov Model Arising in the Slepyan–Palmov Medium

This study investigates the Slepyan–Palmov (SP) model, which describes plane longitudinal waves propagating within a medium comprising a carrier medium and nonlinear oscillators. The primary objective is to analyze the integrability properties of this model. The research entails two key aspects. Firstly, the study explores the group invariant solution by utilizing reductions in symmetry subalgebras based on the optimal system. Secondly, the conservation laws are studied using the homotopy operator, which offers advantages over the conventional multiplier approach, especially when arbitrary functions are absent from both the equation and characteristics. This method proves advantageous in handling complex multipliers and yields significant outcomes.

An improved homotopy perturbation method for dynamic force reconstruction

In many engineering application, it is still a challenge in directly obtaining external force acting on structures. Moreover, many researchers attach more attention to indirect measurement technology using some response signals. This paper presents an improved homotopy perturbation method (IHPM) for dynamic load identification. This method can identify multi-source dynamic sine and triangle force. Firstly, the present method is proposed based on improved homotopy operator. Secondly, the sequence that the present method generates is proved to be a Cauchy sequence when ‖Aeα−I‖<1. Finally, the proposed method is applied to the dynamic load identification of hanging basket structure. Numerical simulations verify the stability and effectiveness of the newly developed method, and also validate that it is superior to the traditional homotopy perturbation method (HPM) and Tikhonov regularization method.

Inequalities for integral operators in Hölder–Morrey spaces on differential forms

The Hölder–Morrey spaces Lambda _{kappa}^{p,tau}(Omega ,wedge ^{l}) are proposed in this paper. The imbedding inequalities for homotopy operator are derived in Hölder–Morrey spaces on differential forms. The Hölder continuity for Riesz potential with envelope function is deduced. As application, some composite theorems, which are associated with conjugate A-harmonic equations on differential forms, are given.

New conservation laws of the Boussinesq and generalized Kadomtsev–Petviashvili equations via homotopy operator

According to the tools of linear algebra and calculus of variations, the conservation laws of Boussinesq and generalized Kadomtsev–Petviashvili (gKP) equations are investigated using multipliers and scaling methods. Using the Euler–Lagrange operator, the determining equations are calculated to find the multipliers of Boussinesq equation and the actual density of gKP equation. Then, the flux and density pairs of Boussinesq equation and the corresponding flux of gKP equation are obtained through 2-dimensional homotopy operator.

Lie algebras with differential operators of any weights

&lt;abstract&gt;&lt;p&gt;In this paper, we define a cohomology theory for differential Lie algebras of any weight. As applications of the cohomology, we study abelian extensions and formal deformations of differential Lie algebras of any weight. Finally, we consider homotopy differential operators on $ \mathrm{L}_{\infty} $ algebras and 2-differential operators of any weight on Lie 2-algebras, and we prove that the category of 2-term $ \mathrm{L}_{\infty} $ algebras with homotopy differential operators of any weight is same as the category of Lie 2-algebras with 2-differential operators of any weight.&lt;/p&gt;&lt;/abstract&gt;

Symbolic-numeric integration of univariate expressions based on sparse regression

The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules.

H-space and loop space structures for intermediate curvatures

For dimensions [Formula: see text] and [Formula: see text], we show that the space of metrics of [Formula: see text]-positive Ricci curvature on the sphere [Formula: see text] has the structure of an [Formula: see text]-space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May, that the path component of this space containing the round metric is weakly homotopy equivalent to an [Formula: see text]-fold loop space.

A noncommutative calculus on the cyclic dual of Ext

We show that if the cochain complex computing Ext groups (in the category of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products.

Symmetry group analysis and conservation laws of the potential modified KdV equation using the scaling method

In this paper, using the Lie symmetry method, we obtain the Lie symmetry group of the potential modified Korteweg–de Vries (potential mKdV) equation, and we construct the optimal system of one-dimensional subalgebras for the Lie symmetry algebra of this equation. Furthermore, similarity reductions related to the infinitesimal generators of the Lie symmetry group are obtained. Also, we construct new conservation laws for this equation by the scaling method. This method is algorithmic and based on the linear algebra tools and calculus of variations. The density of the conservation laws is constructed using the Euler operator and scaling symmetry of the equation and also the associated flux is calculated by the homotopy operator. This density-flux pair renders a conservation law for the potential mKdV equation. We specify the density-flux pairs of conservation laws in ranks 4, 8, 12, 16 and 20 for the potential mKdV equation.

The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

Scaling Symmetry and a New Conservation Law of the Harry Dym Equation

In this paper, we obtain a new conservation law for the Harry Dym equation by using the scaling method. This method is algorithmic and based on variational calculus and linear algebra. In this method, the density of the conservation law is constructed by considering the scaling symmetry of the equation and the associated flux is obtained by the homotopy operator. This density-flux pair gives a conservation law for the equation. A conservation law of rank 7 is constructed for the Harry Dym equation.

Conservation laws, optical molecules, modulation instability and Painlevé analysis for the Chen–Lee–Liu model

In this paper, we will compute the conservation laws (CLs) of Chen–Lee–Liu equation (CLLE) with the help of scaling invariance technique. We will use Euler and Homotopy operators for the evaluation of conserved densities and fluxes. We will also obtain optical dromions with the help of Unified method (UM). By using this method, we will get domain walls, solitary wave and elliptic wave solutions. Moreover, we will investigate the stability as well as the integrability of the governing model by using linear stability technique and Painleve analysis respectively.

Soliton solutions, Painleve analysis and conservation laws for a nonlinear evolution equation

In this paper, we investigate a reputed nonlinear partial differential equation (NLPDE) known as geophysical Korteweg-de Vries (GPKdV) equation. We implement a renowned Unified method (UM) of nonlinear (NL) sciences for the extraction of polynomial and rational function solutions of GPKdV equation, which degenarate to various wave solutions like solitary, soliton (dromions) and elliptic wave solutions. Further more, for the analysis of the integrability of our governing model, we apply Painlevé (P) algorithm to check the singularities structure of the model. The fulfillment of all the requirements of the P test indicates the solvability of the governing equation with the help of inverse scattering transformation (IST) or some linear techniques. Moreover, we calculate conservation laws (CLs) in polynomial form as conserved fluxes and densities by implementing dilation symmetry. We utilize Euler and Homotopy operators for the evaluation of the intended conserved quantities.

Geometrical modeling and numerical analysis of edge dislocation

In this study, we conduct modeling and numerical analysis of a straight edge dislocation in a three-dimensional geometrically nonlinear elastic medium. Based on the fundamental framework of geometrical elasto-plasticity, kinematics of the continuum is represented by the reference R, intermediate B and current S states and local deformations are described by the multiplicative decomposition of the deformation gradient; F = Fe·Fp. Here, the reference and current states are Euclidean submanifolds while the intermediate state B is represented by Weitzenbock manifold with non-zero torsion in the affine connection. Following to the equivalence of torsion and dislocation density tensor through Hodge star operation, a plastic deformation gradient Fp is obtained from the Cartan first structure equation using the homotopy operator. The current state S is determined so that it minimizes the strain energy functional. We solve the variational problem by using isogeometric analysis; Galerkin method with non-uniform B-spline basis function. The present analysis shows that all stress components agree quantitatively well with those of the Volterra dislocation model. Around the dislocation core, stress fields are non-singular on the contrary to the Volterra dislocation. In addition, we found that all stress components distribute asymmetrically due to the geometrical nonlinearity of the model.

Geometric decomposition, potential-based representation and integrability of non-linear systems

Abstract This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed decomposition.

The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds.

Weighted Sobolev $$L^{p}$$ Estimates for Homotopy Operators on Strictly Pseudoconvex Domains with $$C^{2}$$ boundary

We derive estimates in a weighted Sobolev space $$W^{k,p}_{\beta }(D)$$ for a homotopy operator on a bounded strictly pseudoconvex domain D of $$C^2$$ boundary in $${\mathbb {C}}^n$$ . As a result, we show that given any $$2n< p < \infty $$ , $$k > 1$$ , $$q \ge 1$$ , and a $$\overline{\partial }$$ -closed (0, q) form $$\varphi $$ of class $$W^{k,p}(D)$$ , there exists a solution u to $$\overline{\partial } u = \varphi $$ , such that $$u \in W^{k,p}_{\frac{1}{2} - \epsilon }(D)$$ for any $$\epsilon > 0$$ . If $$k=1$$ , then we can take p to be any value between 1 and $$\infty $$ . In other words, the solution gains almost $$\frac{1}{2}$$ -derivative in a suitable sense.

Conserved densities and fluxes for nonlinear Schrödinger equations using scaling invariance approach

We have adopted a direct method for the computation of polynomial conservation laws (CLs) of three nonlinear Schrödinger equations (NLSEs). The equations under consideration are firstly converted to evolution forms. Instead of using advanced differential-geometric tools, our method utilizes tools from linear algebra and variational calculus. This method can be implemented on NLSEs which occur in quantum physics, plasma physics, and fluid dynamics. In case of NLSEs with parameters, our method evaluates conditions on the parameters involved in order to find a sequence of conserved densities. The complete integrability of a NLSE can be predicted by the existence of a large number of CLs of the equation. The method utilizes linear combinations of scaling homogeneous terms having undetermined coefficients for the computation of conserved densities. The undetermined coefficients are being evaluated with the help of variational derivative (Euler operator) while Homotopy operator assists in the determination of conserved fluxes.