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In this work, we continue the development of methods for constructing Lax pairs and recursion operators for nonlinear integrable hyperbolic equations of soliton type, previously proposed in the work of Habibullin et al (2016 J. Phys. A: Math. Theor. 49 035202). This approach is based on the use of the well-known theory of Laplace transforms. The article completes the proof that for any known integrable equation of sine-Gordon type, the sequence of Laplace transforms associated with its linearization admits a third-order finite-field reduction. It is shown that the found reductions are closely related to the Lax pair and recursion operators for both characteristic directions of the given hyperbolic equation. Previously unknown Lax pairs and recursion operators were constructed. [ABSTRACT FROM AUTHOR]

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems. [ABSTRACT FROM AUTHOR]

We study the D-consistency property for negative symmetries of KdV-type equations. Its connection with the D consistency of discrete equations is explained. [ABSTRACT FROM AUTHOR]

We investigate the gauge transformations of the noncommutative modified KP hierarchy and the noncommutative constrained modified KP hierarchy in the Kupershmidt–Kiso version. By introducing the Orlov–Schulman operator, which depends on time variables and dressing operators, we construct the quantum torus symmetry for the noncommutative modified KP hierarchy. We also give a recursion operator for the noncommutative constrained modified KP hierarchy. We present an extended noncommutative modified KP hierarchy. The compatibility equations between the noncommutative modified KP flows and the extended noncommutative modified KP flows are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new integrable fractional Fokas--Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the Riemann--Hilbert approach. Specifically, we focus on the case of the reflectionless potential with a simple pole for the zero boundary condition. And we provide the fractional N-soliton solution in determinant form. In addition, we prove the fractional one-soliton solution rigorously. Notably, we demonstrate that as |t| → ∞, the fractional N-soliton solution can be expressed as a linear combination of N fractional single-soliton solutions. [ABSTRACT FROM AUTHOR]

We discuss the integration of the Cauchy problem for the negative-order nonlinear Schrödinger equation in the class of rapidly decreasing functions via the inverse scattering problem method. In particular, we obtain the time dependence of scattering data of the Zakharov–Shabat system with the potential that is a solution of the considered problem. We give an explicit representation of the one-soliton solution of the negative-order nonlinear Schrödinger equation based on the obtained results. [ABSTRACT FROM AUTHOR]

The Benjamin–Ono equation involving the Hilbert transformation has been studied extensively from different standpoints. Its variant forms and multi-component extensions have been proposed. In this paper, we study the classification of two-component Benjamin–Ono-type systems of the general form. Our classification is carried out by developing the perturbative symmetry approach due to Mikhailov and Novikov. As a result, new two-component integrable Benjamin–Ono type systems are obtained. [ABSTRACT FROM AUTHOR]

It is well known that the Laplace cascade method is an effective tool for constructing solutions to linear equations of hyperbolic type, as well as nonlinear equations of the Liouville type. The connection between the Laplace method and soliton equations of hyperbolic type remains less studied. The article shows that the Laplace cascade also has important applications in the theory of hyperbolic equations of the soliton type. Laplace's method provides a simple way to construct such fundamental objects related to integrability theory as the recursion operator, the Lax pair and Dubrovin-type equations, allowing one to find algebro-geometric solutions. As an application of this approach, previously unknown recursion operators and Lax pairs are found for two nonlinear integrable equations of the sine-Gordon type. [ABSTRACT FROM AUTHOR]

In this work, we consider a new type for the space-like spherical magnetic fibers in 2-dimensional de Sitter space. We first find the normalization for Lorentz force of spherical vector fields, along with various types for magnetic curves in 2-dimensional de Sitter space. Then, we research recursion operators for these spherical vector fields. Also, we give some characterizations of the Lorentz force of the spherical vector fields with normalization and recursion operators. Finally, we have magnetically normalized intensity with a visual ferromagnetic model of optical spherical magnetic curves. [ABSTRACT FROM AUTHOR]

In this article, we introduce optical recursional ferromagnetic model for timelike S N -magnetic fiber in deSitter space. Also, the concept of the recursion operator for timelike S N -magnetic fiber is investigated by using normalized operator. Thus, we obtain recursion total phase for Lorentz forces by using recursion ferromagnetic anholonomy density in deSitter space. Phononic recursion ferromagnetic anholonomy density and sonic magnetic crystals are simulated by distinct geometrical applications in deSitter space. [ABSTRACT FROM AUTHOR]

A partial case of the Guéant–Pu model of options pricing, which takes into account transaction costs and the impact of operations on the market in the absence of a permanent market impact, is considered. Three recursion operators for the considered equation are found using P. Olver's criterion. An hierarchy of generalized symmetries for the model is constructed using the recursion operators. [ABSTRACT FROM AUTHOR]

For a supersymmetric generalized nonlinear Schrödinger (sgNLS) equation, a prolongation algebra L is determined via the super-version of Wahlquist–Estabrook prolongation method. L is successfully realized in terms of the simple Lie superalgebra A (1 , 1) , and a linear spectral problem is established for the sgNLS equation. Based on the linear spectral problem, a recursion operator is derived, and its factorization yields a Hamiltonian operator. These properties for sgNLS equation reduce to the properties of the case B supersymmetric NLS equation of Roelofs and Kersten (J Math Phys 33:2185–2206, 1992), so that the integrability of the latter is better understood. Moreover, an answer is provided to an open question addressed by Roelofs and Kersten. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear stability of solitons and multi-solitons for the following Hirota equation i u t + u x x + 2 | u | 2 u + i α (u x x x + 6 | u | 2 u x) = 0 , (t , x) ∈ R 2 , α ∈ R. Firstly, by employing the variational approach with suitable conservation laws, we establish the orbital stability of the Hasimoto solitons in Sobolev space H 1 (R). Secondly, by choosing a suitable Lyapunov functional and investigating the spectrum of second variation operator around the solitons, we derive that the Hasimoto solitons are also orbitally stable in H m (R) with m ∈ N for any α ∈ R. Finally, pure N -solitons of the Hirota equation are shown to be dynamically stable in H N (R). By constructing a suitable Lyapunov functional, it is found that the multi-solitons are non-isolated constrained minimizers satisfying a 2 N order elliptic equation, and the dynamical stability issue is reduced to study of the spectrum of explicit linearized systems. Our approach in the spectral analysis consists in an invariant for the multi-solitons and new operator identities motivated by recursion operators of the Hirota equation. As a direct consequence, we recover a new proof of the stability of N -solitons for the nonlinear Schrödinger equation [24]. [ABSTRACT FROM AUTHOR]

This article investigates the computational complex wave solutions of the modified Korteweg–de Vries equation combined with an adverse order of the Korteweg–de Vries model. This model was derived in 2017, where the recursion and inverse recursion operators are employed to select the integrable merged MKdV with a negative MKdV model. This integrable property is tested utilizing the Painlevé property. Verosky gave the description and properties of the opposing order recursion operator. We handle this model by implementing eleven contemporary techniques. We obtain a novel formula of complex solitary wave solutions for this model. Complex solitary wave solutions describe wave propagation, and it is also considered more mathematically concise tools to explain more details about the physical properties of models. The main goals of our paper are a comparison between these methods and introducing a novel modified method. All solutions are checked for accuracy by putting them back into the model via two different software (Maple 17 and Mathematica 12). [ABSTRACT FROM AUTHOR]

This study aims to explore the connection between integrable couplings and three-dimensional unital algebras. Expansions in these algebras generate integrable couplings, including nonlinear integrable couplings, and their corresponding Lax pairs and hereditary recursion operators possess specific structures, associated with non-semisimple Lie algebras. Applications to the KdV equation are explicitly presented. • Integrable couplings. • Three dimensional unital algebras. • Recursion operators associated with non-semi-simple matrix Lie algebras. [ABSTRACT FROM AUTHOR]

We study the Hamiltonian dynamics of a spaceship in the background of Alcubierre and Gödel metrics. We derive the Hamiltonian vector fields governing the system evolution, and construct and discuss the associated recursion operators generating the constants of motion. We also characterize relevant master symmetries. [ABSTRACT FROM AUTHOR]

Properties associated with the integrability of the four-dimensional equation of universal hierarchy are considered. In particular, the structure of the algebra of its local symmetries is studied. We show that the second group of exotic cohomologies of this algebra is nontrivial. We prove that the spectral parameter in the well-known covering of this equation is irremovable. A shadow of a nonlocal symmetry was found; using it, we construct the recursion operator. The action of the recursion operator on some local symmetries generates new non-Abelian coverings of the equation. [ABSTRACT FROM AUTHOR]

We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute the corresponding deformed recursion operator. Besides, using the Hamiltonian–Jacobi separability, we construct recursion operators for Hamiltonian vector fields in conformable Poisson–Schwarzschild and Friedmann–Lemaître–Robertson–Walker (FLRW) manifolds, and derive the related constants of motion, Christoffel symbols, components of Riemann and Ricci tensors, Ricci constant and components of Einstein tensor. We highlight the existence of a hierarchy of bi-Hamiltonian structures in both the manifolds, and compute a family of recursion operators and master symmetries generating the constants of motion. [ABSTRACT FROM AUTHOR]

Starting from a local matrix -problem, we successfully use the -dressing method to derive a hierarchy of nonlinear evolution equations including the nonlinear Schrödinger equation as , the vector modified Korteweg–de Vries equation as , and the Lakshmanan–Porsezian–Danielvia equation as via introducing a suitable recursion operator . In addition, we employ the -dressing method to find the -soliton solutions of the vmKdV equation. Finally, the effects of each parameter on interactions between solitons are discussed, and the effects of the characteristic lines on the relative position of the waves are also analyzed. The method for controlling the propagation direction is presented in detail. [ABSTRACT FROM AUTHOR]

In this paper, we consider symmetry and isomonodromy properties of universal special solutions of integrable nonlinear evolution equations, which are analogs of standard wave catastrophe functions known for linear problems. We perform a comparative analysis of two different approaches to the choice of symmetries of integrable nonlinear equations, which can be applied to the study of such special solutions. Some examples are also presented. [ABSTRACT FROM AUTHOR]

We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semidiscrete variant). First, we linearize the equation around its arbitrary solution u. Then we construct a differential (respectively, difference) equation compatible with the linearized equation for any choice of u. This equation defines a surface called a generalized invariant manifold. In a sense, the manifold generalizes the symmetry, which is also a solution to the linearized equation. In this paper, we concentrate on continuous and discrete models of hyperbolic type. It is known that such kinds of equations have two hierarchies of symmetries, corresponding to the characteristic directions. We have shown that a properly chosen generalized invariant manifold allows one to construct recursion operators that generate these symmetries. It is surprising that both recursion operators are related to different parametrizations of the same invariant manifold. Therefore, knowing one of the recursion operators for the hyperbolic type integrable equation (having no pseudo-constants) we can immediately find the second one. [ABSTRACT FROM AUTHOR]

An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a general formula that produces a pre-symplectic operator from a non-gradient adjoint-symmetry. These formulas are illustrated by several examples of linear PDEs and integrable nonlinear PDEs. Additionally, a classification of quasilinear second-order PDEs admitting a multiplicative symmetry recursion operator through the first formula is presented. [ABSTRACT FROM AUTHOR]

Integrals of motion are constructed from noncommutative (NC) Kepler dynamics, generating , , and dynamical symmetry groups. The Hamiltonian vector field is derived in action–angle coordinates, and the existence of a hierarchy of bi-Hamiltonian structures is highlighted. Then, a family of Nijenhuis recursion operators is computed and discussed. [ABSTRACT FROM AUTHOR]

We consider integrability structures of the generalized Hunter–Saxton equation. We obtain the Lax representation with non-removable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give examples of employing the higher order symmetries to constructing exact globally defined solutions for the generalized Hunter–Saxton equation. [ABSTRACT FROM AUTHOR]

We explain the details of the algebraic constructions on nontrivial examples of the mKdV equations related to the and Kac–Moody algebras. Several types of recursion operators appear naturally in formulating the equations and their Hamiltonian structures. We next introduce the resolvent of the Lax operator and demonstrate that it generates the hierarchy of the Lax representations and also the hierarchy of conservation laws of these equations. [ABSTRACT FROM AUTHOR]

In this paper, starting from a spectral problem, we construct a (2 + 1) -dimensional integrable system of nonlinear evolution equations. Based on the Lax pair, the recursion operator and Darboux transformation for the whole hierarchy were constructed. As an application, some exact solutions for the hierarchy are obtained by using the Darboux transformation. [ABSTRACT FROM AUTHOR]

We construct three nonequivalent gradings in the algebra . The first is the standard grading obtained with the Coxeter automorphism using its dihedral realization. In the second, we use , where is the mirror automorphism. The third is , where is the external automorphism of order 3. For each of these gradings, we construct a basis in the corresponding linear subspaces , the orbits of the Coxeter automorphisms, and the related Lax pairs generating the corresponding modified Korteweg–de Vries (mKdV) hierarchies. We find compact expressions for each of the hierarchies in terms of recursion operators. Finally, we write the first nontrivial mKdV equations and their Hamiltonians in explicit form. For , these are in fact two mKdV systems because the exponent 3 has the multiplicity two in this case. Each of these mKdV systems consists of four equations of third order in . For , we have a system of three equations of third order in . For , we have a system of two equations of fifth order in . [ABSTRACT FROM AUTHOR]

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. [ABSTRACT FROM AUTHOR]

We study integrability –in the sense of admitting recursion operators– of two nonlinear equations which are known to possess compacton solutions: the K(m, n) equation introduced by Rosenau and Hyman Dt(u) +Dx(um) + D3x(un) = 0, and the CS Sequation introduced by Coooper, Shepard, and Sodano, Dt(u) + ul−2 Dx(u) + αpDx(up−1u2x) + 2αD2x(upux) = 0. We obtain a full classification of integrable K(m,n)and CSS equations; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of CSS. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of nonlinear equations on a square graph related to the affine algebra A(1)1. This system is the simplest representative of the class of discrete systems corresponding to affine Lie algebras. We find the Lax representation and construct hierarchies of higher symmetries. In neighborhoods of singular points ⋋ = 0 and ⋋ = ∞, we construct formal asymptotic expansions of eigenfunctions of the Lax pair and, based on these expansions, find series of local conservation laws for the system considered. [ABSTRACT FROM AUTHOR]

Abstract An extended equation of dispersive water waves, proposed by Kupershmidt, is considered. By an ansatz on eigenfunctions of its recursion operator, a linear spectral problem, which turns to be type of the energy dependent Schrödinger, is constructed for the system. As by-products, modified systems are presented. Furthermore, a new bi-Hamiltonian system is obtained and shown to be related to a three component generalization of the Camassa–Holm equation under a Miura-type transformation. [ABSTRACT FROM AUTHOR]

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory. [ABSTRACT FROM AUTHOR]

We introduce the general structures of Lie-admissible algebras in the spaces of Gâteaux differentiable operators and establish their connection with the symmetries of operator equations and the mechanics of infinite-dimensional systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose an integrable fractional derivative nonlinear Schrödinger (fDNLS) equation with the aid of the completeness of the squared eigenfunctions for the Kaup–Newell system. Then we further construct an appropriate Riemann–Hilbert problem, from which we obtain a reconstruction formula of the solution of the fDNLS equation. The fractional N -soliton solution is carried out explicitly by means of determinants and the fractional one-soliton solution is verified rigorously. • We first propose the integrable fractional equation in Kaup-Newell system. • The integrable fractional derivative nonlinear Schrödinger equation is derived. • The fractional rational solution is obtained by the limit technique. • We provide a detailed and rigorous proof of the fractional one-soliton solution. [ABSTRACT FROM AUTHOR]

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order α∈(0,1)∪(1,2). It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.

Abstract We consider (3+1)-dimensional second-order evolutionary PDEs where the unknown u enters only in the form of the 2nd-order partial derivatives. For such equations which possess a Lagrangian, we show that all of them have the symplectic Monge–Ampère form and determine their Lagrangians. We develop a calculus for transforming the symmetry condition to the "skew-factorized" form from which we immediately extract Lax pairs and recursion relations for symmetries, thus showing that all such equations are integrable in the traditional sense. We convert these equations together with their Lagrangians to a two-component form and obtain recursion operators in a 2 × 2 matrix form. We transform our equations from Lagrangian to Hamiltonian form by using the Dirac's theory of constraints. Composing recursion operators with the Hamiltonian operators we obtain the second Hamiltonian form of our systems, thus showing that they are bi-Hamiltonian systems integrable in the sense of Magri. By this approach, we obtain five new bi-Hamiltonian multi-parameter systems in (3+1) dimensions. [ABSTRACT FROM AUTHOR]

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L1−1 L2, where the linear differential operators L1 and L2 are chosen such that the ordinary differential equation (L2 −λL1)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L2, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L1U˜ = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution U˜ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations. [ABSTRACT FROM AUTHOR]

We continue the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation uty = uxuxy − uyuxx, (2) the Pavlov equation uyy = utx + uyuxx − uxuxy, (3) the universal hierarchy equation uyy = utuxy − uyutx, and (4) the modified Veronese web equation uty = utuxy − uyutx. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries. [ABSTRACT FROM AUTHOR]

We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation. [ABSTRACT FROM AUTHOR]