Nonlocal Symmetries Research Articles

Bäcklund transformations, nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the nth Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type N-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.

Non-local symmetry, interaction solutions and conservation lawsof the $$(1+1)$$-dimensional Wu–Zhang equation

Non-local symmetry, interaction solutions and conservation lawsof the $$(1+1)$$-dimensional Wu–Zhang equation

Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

Applications of the q-Srivastava-Attiya Operator Involving a Certain Family of Bi-Univalent Functions Associated with the Horadam Polynomials

In this article, by making use of the q-Srivastava-Attiya operator, we introduce and investigate a new family SWΣ(δ,γ,λ,s,t,q,r) of normalized holomorphic and bi-univalent functions in the open unit disk U, which are associated with the Bazilevič functions and the λ-pseudo-starlike functions as well as the Horadam polynomials. We estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to the holomorphic and bi-univalent function class, which we introduce here. Furthermore, we establish the Fekete-Szegö inequality for functions in the family SWΣ(δ,γ,λ,s,t,q,r). Relevant connections of some of the special cases of the main results with those in several earlier works are also pointed out. Our usage here of the basic or quantum (or q-) extension of the familiar Hurwitz-Lerch zeta function Φ(z,s,a) is justified by the fact that several members of this family of zeta functions possess properties with local or non-local symmetries. Our study of the applications of such quantum (or q-) extensions in this paper is also motivated by the symmetric nature of quantum calculus itself.

Fuzzy Differential Subordinations Based upon the Mittag-Leffler Type Borel Distribution

In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series Bλ,α,β(z) of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function Eα,β(z). Using the above-mentioned operator Bλ,α,β, we also introduce and study a class Mλ,α,βFη of holomorphic and univalent functions in the open unit disk Δ. The Mittag-Leffler-type functions, which we have used in the present investigation, belong to the significantly wider family of the Fox-Wright function pΨq(z), whose p numerator parameters and q denominator parameters possess a kind of symmetry behavior in the sense that it remains invariant (or unchanged) when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. Here, in this article, we have used such special functions in our study of a general Borel-type probability distribution, which may be symmetric or asymmetric. As symmetry is generally present in most works involving fuzzy sets and fuzzy systems, our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetry features.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.

Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. II: Connections with the conservation law method

In this paper, we show direct connections between the conservation law (CL)-based method and the differential invariant (DI)-based method for obtaining nonlocally related systems and nonlocal symmetries for a given partial differential equation (PDE) system. For a PDE system with two independent variables, we show that the CL method is a special case for the DI method. For a PDE system with at least three independent variables, we show that the CL method, for a curl-type CL, is a special case for the DI method. We also consider the situation for a self-adjoint, i.e. variational, linear PDE system. Here, a solution of the linear PDE system yields a nonlocally related system for both approaches. In particular, the resulting nonlocally related systems need not be invertibly equivalent. Through an example, we show that three distinct nonlocally related systems can be obtained from an admitted point symmetry.

Discrete multi-soliton solutions and dynamics for a reverse-time nonlocal nonlinear self-dual network equation

Inspired by the idea of Ablowitz and Musslimani, a reverse-time nonlocal nonlinear self-dual network equation is proposed under certain nonlocal symmetry reduction. N-fold Darboux transformation technique is used to construct discrete unstable and stable multi-soliton solutions in zero seed background for this system. Based on the asymptotic and graphic analysis, structures of one-, two- and three-soliton solutions are shown graphically. It is found that the individual components in this nonlocal equation show unstable temporal soliton structures whereas the combined potential terms exhibit stable soliton structures. Dynamical behaviors for one-soliton solutions are investigated via numerical simulations. It is worth noting that the solutions of this nonlocal equation are clearly different from those of its corresponding local part. Results obtained in this paper may be helpful for the explanation of the propagation of electrical signals.

Nonlocal symmetries and molecule structures of the KdV hierarchy

The nonlocal symmetries are important because they carry information about the existence of Darboux–Backlund and linearizing transformations, and they also allow us to construct nontrivial solutions to systems. In this paper, we focus on investigating three sets of nonlocal symmetries which are realized as appropriate local symmetries of related auxiliary systems for the Korteweg–de Vries hierarchy. Specifically, we construct infinitely many nonlocal symmetries from three different aspects using residues, Lax pairs and quadratic pseudopotentials for each member of the Korteweg–de Vries hierarchy and show how these nonlocal symmetries connect each other. These infinitely many nonlocal symmetries enable us to construct multiple soliton solutions. Moreover, we adapt the multiple soliton solutions derived from nonlocal symmetries and describe a procedure by using velocity resonance mechanism to find molecule solutions, which can be found not only in the optical systems, but also in fluid systems, for the combined third-fifth-order Korteweg–de Vries equation. Several types of molecule structures including soliton molecules, breather molecules and soliton–breather molecules are illustrated.

Twisted symmetric trilayer graphene: Single-particle and many-body Hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field

We derive the Hamiltonian for trilayer moir\'e systems with the Coulomb interaction projected onto the bands near the charge neutrality point. Motivated by the latest experimental results, we focus on the twisted symmetric trilayer graphene (TSTG) with a mirror-symmetry with respect to the middle layer. We provide a full symmetry analysis of the non-interacting Hamiltonian with a perpendicular displacement field coupling the band structure made otherwise of the twisted bilayer graphene (TBG) and the high velocity Dirac fermions, and we identify a hidden non-local symmetry of the problem. In the presence of this displacement field, we construct an approximate single-particle model, akin to the tripod model for TBG, capturing the essence of non-interacting TSTG. We also derive more quantitative perturbation schemes for the low-energy physics of TSTG with displacement field, obtaining the corresponding eigenstates. This allows us to obtain the Coulomb interaction Hamiltonian projected in the active band TSTG wavefunctions and derive the full many-body Hamiltonian of the system. We also provide an efficient parameterization of the interacting Hamiltonian. Finally, we show that the discrete symmetries at the single-particle level promote the $\mathrm{U} \left( 2 \right) \times \mathrm{U} \left( 2 \right)$ spin-valley symmetry to enlarged symmetry groups of the interacting problem under different limits. The interacting part of the Hamiltonian exhibits a large $\mathrm{U} \left( 4 \right) \times \mathrm{U} \left( 4 \right) \times \mathrm{U} \left( 4 \right) \times \mathrm{U} \left( 4 \right)$ symmetry in the chiral limit. Moreover, by identifying a new symmetry which we dub spatial many-body charge conjugation, we show that the physics of TSTG is symmetric around charge neutrality.

Localization of nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

The nonlocal symmetry of the Sawada–Kotera (SK) equation is constructed with the known Lax pair. By introducing suitable and simple auxiliary variables, the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further. On the other hand, the group invariant solutions of the SK equation are constructed with the classic Lie group method. In particular, by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.

Some generalized isospectral-nonisospectral integrable hierarchies

In this article, with the aid of the Lie algebra A1 composed of second order matrices and Lie algebra A2 composed of third order matrices, some new soliton hierarchies of evolution equations are deduced and the corresponding Hamiltonian structures are also worked out by utilizing the trace identity. Specially, one of the integrable soliton hierarchicy is reduced to the generalized Fokker–Plank equation (gFP) and special bond pricing equation. Next, the nonlinear self-adjointness of the generalized Fokker–Plank equation is verified and conservation laws are constructed with the aid of Ibragimov’ method. Moreover, we investigate the coverings and the nonlocal symmetries of the generalized Fokker–Plank equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product. Besides, we apply the Li’s method to obtain two basic Darboux transformations of (gFP) with two different forms of T.

Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation

Nonlocal symmetries related to the Bäcklund transformation (BT) for the modified KdV-sine-Gordon (mKdV-SG) equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal transformations. Then through the parameter expansion procedure, an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are derived. Finally, several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.

Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies

This paper aims to explore nonlocal complex reverse-spacetime modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems and to construct their soliton solutions by the inverse scattering transforms. The corresponding inverse scattering problems are formulated by building the associated Riemann-Hilbert problems. A formulation of solutions to specific Riemann-Hilbert problems, with the jump matrix being the identity matrix, is established, where eigenvalues could equal adjoint eigenvalues, and thus N-soliton solutions to the nonlocal complex reverse-spacetime mKdV hierarchies are obtained from the reflectionless transforms.

On general solitons in the parity-time-symmetric defocusing nonlinear Schrödinger equation

Via the Hirota bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method, two types of solitons, namely general solitons on a background of periodic waves and rational soliton solutions, to the parity-time-symmetric nonlocal nonlinear Schrodinger equation with the defocusing-type nonlinearity for nonzero boundary condition are investigated. These two types of soliton solutions are constructed by constraining the tau functions of single-component KP hierarchy satisfying the dimension reduction and the nonlocal symmetry and the complex conjugated conditions. For the solitons on a background of periodic waves, we first construct the periodic solution to provide the background of periodic waves and then combine the periodic solution with general soliton solution, which generate solitons on a background of periodic waves. For the rational solitons, we mainly investigate the dynamical behaviours of interactions between several individual rational dark–antidark solitons, antidark–antidark solitons, and antidark–dark solitons, which are elastic collisions with no phase shift. The degenerated soliton cases are also discussed, in which only one-antidark soliton survives.

Nonlocal symmetries of some nonlinear partial differential equations with third-order Lax pairs

Nonlocal symmetries of some nonlinear partial differential equations with third-order Lax pairs

Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation

On the algebra of nonlocal symmetries for the 4D Martínez Alonso–Shabat equation

We consider the 4D Martínez Alonso–Shabat equation Euty=uzuxy−uyuxz (also referred to as the universal hierarchy equation) and using its known Lax pair construct two infinite-dimensional differential coverings over 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 Morozov and Sergyeyev (2014) and contain the constructed there infinite hierarchy of commuting symmetries as a subalgebra in a much bigger Lie algebra.

Infinitely many commuting nonlocal symmetries for modified Martínez Alonso–Shabat equation

We study the modified Martínez Alonso–Shabat equationuyuxz+αuxuty−(uz+αut)uxy=0and present its recursion operator and an infinite commuting hierarchy of full-fledged nonlocal symmetries. To date such hierarchies were found only for very few integrable systems in more than three independent variables.

Nonlocal symmetry of CMA generates ASD Ricci-flat metric with no Killing vectors

The complex Monge–Ampère (CMA) equation in a two-component form is treated as a bi-Hamiltonian system. I present explicitly the first nonlocal symmetry flow in each of the two hierarchies of this system. An invariant solution of the CMA equation with respect to these nonlocal symmetries is constructed, which, being a noninvariant solution in the usual sense, does not undergo symmetry reduction in the number of independent variables. I also construct the corresponding four-dimensional anti-self-dual Ricci-flat metric with either the Euclidean or neutral signature. It admits no Killing vectors, which is one of the characteristic features of the famous gravitational instanton K3. For the metric with the Euclidean signature, relevant for gravitational instantons, I explicitly calculate the Levi-Civita connection one-forms and the Riemann curvature tensor.