We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal symmetries and formulae of nonlocal nonlinear superposition of solutions of these equations were used then for construction of chains of exact solutions. Linearization by means of the Legendre transformations of a second-order PDE with three independent variables allowed to obtain nonlocal superposition formulae for solutions of this equation, and to generate new solutions from group invariant solutions of a linear equation.
In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg--de Vries equation. Lie point symmetries are obtained based on a general theory and used to reduce these equations to nonlocal and local ordinary differential equations separately; namely one symmetry may allow reductions to both nonlocal and local equations depending on how the invariant variables are chosen. For the nonlocal modified Korteweg--de Vries equation, analogously to the…
We construct new infinite hierarchies of nonlocal symmetries and cosymmetries for the Krichever--Novikov equation using the inverse of the fourth-order recursion operator of the latter.
Using the lax pair, nonlocal symmetry of the coupled Hirota equation is obtained. By introducing an appropriate auxiliary dependent variable the nonlocal symmetry is successfully localized to a Lie point symmetry. For the closed prolongation, one-dimensional optimal systems and nonlocal conservation laws of the coupled Hirota equation are studied.
We consider the 4D Mart\'ınez Alonso-Shabat equation $u_{ty} = u_z u_{xy} - u_y u_{xz}$ (also referred to as the universal hierarchy equation) and using its known Lax pair construct two infinite-dimensional differential coverings over $\mathcal{E}$. In these coverings, we give a complete description of the Lie algebras of nonlocal symmetries. In particular, our results generalize the ones obtained in [O.I.Morozov, A.Sergyeyev, The four-dimensional Mart\'ınez Alonso-shabat equation: reductions and nonlocal symmetries. J. of Geom. and Phys. 85 (2014), 40--45 (arXiv:1401.7942v2)] and contain the …
This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $λ$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the $λ$-coverings method. The $λ$-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknow…
Zero curvature formulations, pseudo-potentials, modified versions, ``Miura transformations'', and nonlocal symmetries of the Korteweg--de Vries, Camassa-- Holm and Hunter--Saxton equations are investigated from an unified point of view: these three equations belong to a two--parameters family of equations ``describing pseudo-spherical surfaces'', and therefore their basic integrability properties can be studied by geometrical means.
In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are discussed for the first time. With lax pair of time-dependent coefficient AKNS system, the nonlocal symmetries are obtained, and they are successfully localized to a Lie point symmetries by introducing a suitable auxiliary dependent variable. Furthermore, using the obtained Lie point symmetries of closed system, we give out two types of symmetry reduction and explicit analytic solutions. For some interesting solutions, the figures are given out to show their dynamic behavior.
We construct infinite hierarchies of nonlocal higher symmetries for the oriented associativity equations using solutions of associated vector and scalar spectral problems. The symmetries in question generalize those found by Chen, Kontsevich and Schwarz (arXiv:hep-th/0508221) for the WDVV equations. As a byproduct, we obtain a Darboux-type transformation and a (conditional) Bäcklund transformation for the oriented associativity equations.
We give a general account of nonlocal symmetries in symmetric space models and their relation to the AdS/CFT correspondence. In particular, we study a master symmetry which generates the spectral parameter and acts as a level-raising operator on the classical Yangian generators. The master symmetry extends to an infinite tower of symmetries with nonlocal Casimir elements as associated conserved charges. We discuss the algebraic properties of these symmetries and establish their role in explaining the recently observed one-parameter deformation of holographic Wilson loops. Finally, we provide a…
We consider the recursion operators with nonlocal terms of special form for evolution systems in (1+1) dimensions, and extend them to well-defined operators on the space of nonlocal symmetries associated with the so-called universal Abelian coverings over these systems. The extended recursion operators are shown to leave this space invariant. These results apply, in particular, to the recursion operators of the majority of known today (1+1)-dimensional integrable evolution systems. We also present some related results and describe the extension of them and of the above results to (1+1)-dimensi…
It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all known today recursion operators and are much easier to verify than those found in earlier work. We also give explicit formulas for the nonlocal parts of higher recursion operators, Poisson and symplecti…
We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2,R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and th…
We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced $N^{th}$ order ordinary differential equation (ODE), $N=2, 3,4,...$, in this chain yields $(N-1)^{th}$ order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced $N^{th}$ order ODE, $N=2, 3,4,$, in the Abel chain always ends at the $(N-1)^{th}$ ord…
Starting from nonlocal symmetries related to Bäcklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact form which comes from BT is presented and used to make researches in the following three subjects: two sets of negative pKdV hierarchies and their corresponding bilinear forms are constructed; the nonlocal symmetry is localized by introduction of suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some nov…
A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $λ$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $λ$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In this framework, by embedding $\mathcal{Y}$ into a suitable system $\mathcal{Y}^{\prime}$ determined by the function $λ$, any $λ$-symmetry of $\mathcal{Y}$ can be recovered by a local symmetry of $\mathcal{Y}^{\prime}$. As a consequence, the reduction method of Muriel and Romero f…
A demonstration of how the point symmetries of the Chazy Equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy Equation under a generalized transformation, and find the point symmetries of the Chazy Equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy Equation which is given by a Right Painlevé Series. The singularity analysis is applied to the new third-ord…
Conventionally, factorized scattering in two dimensions is argued to be a consequence of the conservation of local higher charges. However, integrability may well be realized via nonlocal charges, while higher local charges are not known. Here we address the question of whether a nonlocal Yangian symmetry implies factorized scattering of the S-matrix. We explicitly study the constraints on three-particle scattering processes of particles transforming in the fundamental representations of su(N), u(1|1), and the centrally extended su(2|2) underlying the dynamic scattering and hexagon form factor…
A computational classification of contact symmetries and higher-order local symmetries that do not commute with $t,x$, as well as local conserved densities that are not invariant under $t,x$ is carried out for a generalized version of the Krichever-Novikov equation. Several new results are obtained. First, the Krichever-Novikov equation is explicitly shown to have a local conserved density that contains $t,x$. Second, apart from the dilational point symmetries known for special cases of the Krichever-Novikov equation and its generalized version, no other local symmetries with low differential …
We found a new symplectic structure and a recursion operator for the Sasa--Satsuma equation widely used in nonlinear optics, $$ p_t=p_{xxx}+6 p q p_x+3 p (p q)_x,\quad q_t=q_{xxx}+6 p q q_x+3 q (p q)_x, $$ along with an integro-differential substitution linking this system to a third-order generalized symmetry of the complex sine-Gordon II system $$ u_{xy}=\frac{v u_x u_y}{u v + c} + (2 u v + c)(u v + c) k u,\qquad v_{xy}=\frac{u v_x v_y}{u v + c} + (2 u v + c)(u v + c) k v, $$ where $c$ and $k$ are arbitrary constants. Combining these two results yields a highly nonlocal hereditary recursion …
We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction-diffusion systems with nonlocal quadratic nonlinearities and the nonlinear Schrodinger equation with a…
New nonlocal symmetries and conservation laws are derived for Maxwell's equations using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling…
In this work, we study the dynamics of rogue waves in the partially $\cal{PT}$-symmetric nonlocal Davey-Stewartson(DS) systems. Using the Darboux transformation method, general rogue waves in the partially $\cal{PT}$-symmetric nonlocal DS equations are derived. For the partially $\cal{PT}$-symmetric nonlocal DS-I equation, the solutions are obtained and expressed in term of determinants. For the partially $\cal{PT}$-symmetric DS-II equation, the solutions are represented as quasi-Gram determinants. It is shown that the fundamental rogue waves in these two systems are rational solutions which a…
In this paper we give a geometric interpretation of a reduction method based on the so called $λ$-variational symmetry (C. Muriel, J.L. Romero and P. Olver 2006 \emph{Variational $C^{\infty}$-symmetries and Euler-Lagrange equations} J. Differential equations \textbf{222} 164-184). In general this allows only a partial reduction but it is particularly suitable for the reduction of variational ODEs with a lack of computable local symmetries. We show that this method is better understood as a nonlocal symmetry-reduction.
We study point symmetries, the corresponding conserved densities and hierarchies of four new bi-Hamiltonian heavenly systems in 3+1 dimensions which we discovered recently. We exhibit an important role played by the inverse recursion operators in the description of the hierarchies. Their use is twofold, either to provide the correct bi-Hamiltonian representation or to generate nonlocal symmetry flows. Invariant solutions w.r.t. nonlocal symmetries will generate (anti-)self-dual gravitational metrics which do not admit Killing vectors which is a characteristic feature of $K3$ gravitational inst…
Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonl…
Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power onlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrodinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg--de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or…
In classical shallow water wave (SWW) theory, there exist two integrable one-dimensional SWW equation [Hirota-Satsuma (HS) type and Ablowitz-Kaup-Newell-Segur (AKNS) type] in the Boussinesq approximation. In this paper, we mainly focus on the integrable SWW equation of AKNS type. The nonlocal symmetry in form of square spectral function is derived starting from its Lax pair. Infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter. These nonlocal symmetries can be localized and the SWW equation is extended to enlarged system with auxiliary dependent var…
In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show on a few examples, both in partial differential and partial difference equations when this happens. Moreover we show that the infinitesimal generators of generalized symmetries depending on arbitrary functions, both for continuous and discrete equations, effectively play the role of master symmetries.
We apply the theory of infinitesimal transformations for the study of a family of 1+1 fifth-order partial differential equations which have been proposed before for the description of multiple kink solutions. In this analysis we perform a complete classification of the Lie symmetries and of the one-dimensional optimal system. The results are applied for the derivation of similarity solutions and in particular we find travel-wave and scaling solutions. We show that the kink-solution of these equations can be recovered by the use of the Lie symmetries, while new solutions are also derived.
Nonlocally related partial differential equation (PDE) systems are useful in the analysis of a given PDE system. It is known that each local conservation law of a given PDE system systematically yields a nonlocally related system. In this paper, a new and complementary method for constructing nonlocally related systems is introduced. In particular, it is shown that each point symmetry of a given PDE system systematically yields a nonlocally related system. Examples include applications to nonlinear diffusion equations, nonlinear wave equations and nonlinear reaction-diffusion equations. As a c…
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining nonclassical symmetry of a PDE and make it easier to solve the system of nonlinear determining equations. As example, new nonclassical symmetries are exhibited for a class of generalized Burgers equation and KdV-type equations are given.
A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial difference scheme involving N difference equations, where N is the number of independent and dependent variable. We restrict to one scalar function of two independent variables. As examples, invariant discretizations of the heat, Burgers and Korteweg-de Vries equations are presented. Some exact solutions of the discrete schemes are obtained.
In this short note we discuss a new formula for solving the nonlocal $\overline{\partial}$-problem, and discuss application to the Manakov--Zakharov dressing method. We then explicitly apply this formula to solving the complex (2+1)D Kadomtsev--Petviashvili equation and complex (2+1)D completely integrable generalization of the (2+1)D Kaup--Broer (or Kaup--Boussinesq) system. We will also discuss how real (1+1)D solutions are expressed using this formalism. It is simple to express the formalism for finite gap primitive solutions from [10], [8] using the formalism of this note. We also discuss …
We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanli) et al. (arXiv: nlin.SI/0203036) is bihamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators. Using this, we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution, and the evolution systems corresponding to these symmetries are bihamiltonian as well. We also show that upon introduction of suitable nonlocal …
We define rigorously operators of the form $f(\partial_t)$, in which $f$ is an analytic function on a simply connected domain. Our formalism is based on the Borel transform on entire functions of exponential type. We study existence and regularity of real-valued solutions for the nonlocal in time equation \begin{equation*} f(\partial_t) ϕ= J(t) \; \; , \quad t\in \mathbb{R}\; , \end{equation*}. and we find its more general solution as a restriction to $\mathbb{R}$ of an entire function of exponential type. As an important special case, we solve explicitly the linear nonlocal zeta field equatio…
We give a geometrical interpretation of the notion of $μ$-prolongations of vector fields and of the related concept of $μ$-symmetry for partial differential equations (extending to PDEs the notion of $λ$-symmetry for ODEs). We give in particular a result concerning the relationship between $μ$-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local $μ$-symmetries and nonlocal standard symmetries.
For the class of systems of PDEs, for which infinitesimal translations (with respect to some (in)dependent variables) possess specific finite-dimensional invariant subspaces of the space of generalized symmetries of the system considered. We establish when there exist generalized symmetries from these subspaces, which depend from the above-mentioned variables.
A key feature of integrable systems is that they can be solved to obtain exact analytical solutions. We show how new models can be constructed through generalisations of some well known nonlinear partial differential equations with PT-symmetries whilst preserving integrability. Subsequently, we develop new methods from well-known ones to obtain exact analytical soliton solutions for these new systems. The first PT-symmetric generalization we take are extensions to the complex and multicomplex fields. In agreement with the reality property present in PT-symmetric non-Hermitian quantum systems, …
When we consider a differential equation $Δ=0$ whose set of solutions is ${\cal S}_Δ$, a Lie-point exact symmetry of this is a Lie-point invertible transformation $T$ such that $T({\cal S}_Δ)={\cal S}_Δ$, i.e. such that any solution to $Δ=0$ is tranformed into a (generally, different) solution to the same equation; here we define {\it partial} symmetries of $Δ=0$ as Lie-point invertible transformations $T$ such that there is a nonempty subset ${\cal P} \subset {\cal S}_Δ$ such that $T({\cal P}) = {\cal P}$, i.e. such that there is a subset of solutions to $Δ=0$ which are transformed one into t…
Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, \textit{i.e.}, classes of differential equations that are in correspondence with their Lie point symmetries. In particular, we determine hierarchies of second order partial differential equations uniquely characterized by affine transformations of $\mathbb{R}^{n+m}$, and a system of two third order partial differential equations in two independent variables uniquely determined by the Lie algebra of projective transformations of $\mathbb{R}^4$.
Following a recently introduced approach to approximate Lie symmetries of differential equations which is consistent with the principles of perturbative analysis of differential equations containing small terms, we analyze the case of approximate $Q$--conditional symmetries. An application of the method to a hyperbolic variant of a reaction--diffusion--convection equation is presented.
The conservation laws of electromagnetism, and implicitly all theories built from quadratic Lagrangians, are extended to a continuum of nonlocal versions. These are associated with symmetries of a class of equal time field correlation functions and give results for both connected and disconnected branches of the general linear group of the space. It is generally assumed that manifestly covariant Lagrangians are the necessary starting point for physical theories. Here we show that the EOM derived from any of these can also follow from a broad class of nonlocal ones and each generally gives a di…
Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant variables) on specific finite-dimensional subspaces of the space of generalized symmetries of some system of partial differential equations, we explicitly determine the dependance of coefficients of generalized symmetries from these subspaces on the above-mentioned variables. We establish the connection of our results with the theory of quasiexactly solvable models. Some generalizations of the approach proposed also are discussed.
We show that a class of previously defined maps, called self-dual and causal morphisms, form classical symmetries of Yang-Mills fields in four complex dimensions. These maps generalize conformal transformations, and admit a nonlocal pullback connection that preserves the equations of the theory. First it is shown that self-dual morphisms form symmetries of the anti-self-dual Yang-Mills equations under this pullback. Then a supersymmetric generalization of causal morphisms is defined, which preserves solutions of the field equations for N=3 supersymmetric Yang-Mills theory. As a special case, t…
We study the spectral correspondence between a particular class of Schrodinger equations and supersymmetric quantum integrable model (QIM). The latter, a quantized version of the Ablowitz-Kaupp-Newell-Segur (AKNS) hierarchy of nonlinear equations, corresponds to the thermodynamic limit of the Perk-Schultz lattice model. By analyzing the symmetries of the ordinary differential equation (ODE) in the complex plane, it is possible to obtain important objects in the quantum integrable model in exact form, under an exact spectral correspondence. In this manuscript our main interest lies on the set o…
