Nonlocal Symmetries Research Articles

Non-defective degeneracy in non-Hermitian bipartite system

Starting from a Hermitian operator with two distinct eigenvalues, we construct a non-Hermitian bipartite system in Gaussian orthogonal ensemble according to random matrix theory, where we introduce the off-diagonal fluctuations through random eigenkets and realizing the bipartite configuration consisting of two D × D subsystems (with D being the Hilbert space dimension). As required by the global thermalization (chaos), one of the two subsystems is fully ranked, while the other is rank deficient. For the latter (rank-deficient) subsystem, there is a block with non-defective degeneracies that contains non-local symmetries, as well as the accumulation effect of the linear map in adjacent eigenvectors. The maximally mixed state formed by the eigenvectors of this special region does not exhibit thermal ensemble behavior (neither canonical or Gibbs), and displays similar characteristics to the corresponding reduced density, which can be verified through the Loschmidt echo and variance of the imaginary spectrum. This non-defective degeneracy region partially meets the Lemma in 10.1103/PhysRevLett.122.220603 and theorem in 10.1103/PhysRevLett.120.150603. The coexistence of strong entanglement and initial state fidelity in this region make it possible to achieve a maximally mixed density, which, however, does not correspond to a thermal canonical ensemble (with complete insensitivity to the environmental energy or temperature). Outside this region, the collection of eigenstates (reduced density) always exhibit restriction on the corresponding Hilbert space dimension (with, e.g., infinite number of bound states), and thus suppress the global thermalization. There are abundant physics for those densities in Hermitian and non-Hermitian bases, which we investigate separately in this work. For example, the disentangling effect originates from non-Hermitian skin effect where the coherence exists along the direction orthogonal to the entangled boundaries of the Loschmidt echo spectrum in the Hermitian basis, while it originates from the many-body localization with the coherence among echo boundaries in the non-Hermitian basis which is disorder-free.

Nonlocal symmetries of two 2-component equations of Camassa–Holm type

Nonlocal symmetries of two 2-component equations of Camassa–Holm type

Two-component integrable extension of general heavenly equation

We introduce an integrable two-component extension of the general heavenly equation and prove that the solutions of this extension are in one-to-one correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we demonstrate that if the metrics in question are hyper-para-Kähler, then our system reduces to the general heavenly equation. We also present an infinite hierarchy of nonlocal symmetries, as well as a recursion operator, for the system under study.

Conservation laws, nonlocal symmetries, and exact solutions for the Cargo–LeRoux model with perturbed pressure

Conservation laws, nonlocal symmetries, and exact solutions for the Cargo–LeRoux model with perturbed pressure

Interaction solutions for variable coefficient coupled Lakshmanan–Porsezian–Daniel equations via nonlocal residual symmetry and CTE method

Interaction solutions for variable coefficient coupled Lakshmanan–Porsezian–Daniel equations via nonlocal residual symmetry and CTE method

Nonlocal symmetries, soliton-cnoidal wave solution and soliton molecules to a (2+1)-dimensional modified KdV system

A (2+1)-dimensional modified KdV (2DmKdV) system is considered from several perspectives. Firstly, residue symmetry, a type of nonlocal symmetry, and the Bäcklund transformation are obtained via the truncated Painlevé expansion method. Subsequently, the residue symmetry is localized to a Lie point symmetry of a prolonged system, from which the finite transformation group is derived. Secondly, the integrability of the 2DmKdV system is examined under the sense of consistent tanh expansion solvability. Simultaneously, explicit soliton-cnoidal wave solutions are provided. Finally, abundant patterns of soliton molecules are presented by imposing the velocity resonance condition on the multiple-soliton solution.

Nonlocal symmetries and solutions of the multi-dimensional integrable long water wave equations

In this paper, the (2+1)-dimensional integrable long water wave equations (LWWs) are constructed for the first time using the conservation law of the (1+1)-dimensional LWWs. The new (1+1)-dimensional LWWs can be obtained by introducing a constraint to the (2+1)-dimensional LWWs. This new (1+1)-dimensional LWWs are studied by using nonlocal symmetry methods for the first time. The closed system corresponding to nonlocal symmetry is established by the lax pairs of equations and the potential function determined using conservation laws. Exact solutions of the equations are obtained by finite symmetry transformation and symmetry approximation of this closed system. The dynamic behavior of the equations is studied by means of figures of the exact solutions.

The (2+1)-dimensional generalized Benjamin–Ono equation: Nonlocal symmetry, CTE solvability and interaction solutions

In this paper, the nonlocal symmetry of the (2+1)-dimensional generalized Benjamin–Ono equation are obtained by using the truncated Painlevé expansion. The nonlocal symmetry are localized by introducing auxiliary variables. Furthermore, this equation is also proved to be consistent tanh expansion solvable, and three classes of exact solutions are derived.

On Recursion Operators for Full-Fledged Nonlocal Symmetries of the Reduced Quasi-classical Self-dual Yang–Mills Equation

We introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang–Mills equation. It turns out that the discovered recursion operators can be interpreted as infinite-dimensional matrices of differential functions which act on the generating vector functions of the nonlocal symmetries simply by matrix multiplication. To the best of our knowledge, there are no other examples of such recursion operators in the literature so far, so our approach is completely innovative. Further, we investigate the algebraic properties of the discovered operators and discuss the R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}$$\\end{document}-algebra structure on the set of all recursion operators for full-fledged nonlocal symmetries of the equation in question. Finally, we illustrate the action of the obtained recursion operators on particularly chosen full-fledged symmetries and emphasize their advantages compared to the action of traditionally used recursion operators for shadows.

Nonlocal symmetry and group invariant solutions of dissipative (2+1)-dimensional AKNS equation

In this paper, the nonlocal symmetry of dissipative (2+1)-dimensional AKNS equation is constructed by auxiliary equation method. In order to better study new group invariant solutions of AKNS equation with the help of nonlocal symmetry, we introduce an new auxiliary dependent variable, the (2+1) dimensional AKNS equation is extended to a new closed prolonged system. Therefore, the nonlocal symmetry of the original equation is the Lie point symmetry of the new prolonged system. Then, by solving the initial-value problems, the finite symmetry transformation for the prolonged system is derived. Furthermore, by using symmetry reduction method to the prolonged system, the interaction solution between cnoidal waves and solitary waves is given.

Nonlocal symmetry and exact solutions of the (2+1)-dimensional Gerdjikov–Ivanov equation

Nonlocal symmetry and exact solutions of the (2+1)-dimensional Gerdjikov–Ivanov equation

Nonlocal Symmetries, Consistent Riccati Expansion Solvability and Interaction Solutions of the Generalized Ito Equation

In this paper, we investigate the generalized Ito equation. By using the truncated Painlevé analysis method, we successfully derive its nonlocal symmetry and Bäcklund transformation, respectively. By introducing new dependent variables for the nonlocal symmetry, we find the corresponding Lie point symmetry. Moreover, we construct the interaction solution between soliton and cnoidal periodic wave of the equation by considering the consistent tanh expansion method. The conservation laws of the equation are also obtained with a detailed derivation.

Nonlocal residual symmetries, N-th Bäcklund transformations and exact interaction solutions for a generalized Broer–Kaup–Kupershmidt system

Nonlocal residual symmetries, N-th Bäcklund transformations and exact interaction solutions for a generalized Broer–Kaup–Kupershmidt system

An efficient way to determine Liouvillian first integrals of rational second order ordinary differential equations

The Lie and Darboux methods to obtain Liouvillian first integrals of rational second order ordinary differential equations (rational 2ODEs) are very powerful. Nevertheless, there are cases where these procedures may encounter difficulties. Namely, when the Darboux polynomials present in the integrating factor of the 2ODE have a very high degree and/or when the 2ODE does not admit point symmetries. In [3,4] we developed a method (S-function method) that is successful in treating certain classes of rational second order ordinary differential equations that are particularly ‘resistant’ to canonical Lie methods and to Darbouxian approaches. However, although determining the S-function is in general much more efficient than determining Darboux polynomials or nonlocal symmetries, for very complicated rational 2ODEs, even finding the S-function itself can be quite hard. In this work we present a simple way of (in almost all cases) computing the S-function with a very efficient procedure. New version program summaryProgram Title: InSyDE – Invariants and Symmetries of (rational second order ordinary) Differential Equations.CPC Library link to program files:https://doi.org/10.17632/4ytft6zgk7.2Licensing provisions: CC by NC 3.0Programming language: Maple 17Journal reference of previous version: Comput. Phys. Commun. 234 (2019) 302–314Does the new version supersede the previous version?: Yes.Nature of problem: Determining Liouvillian first integrals of rational second order ordinary differential equations.Solution method: The method is explained in the body of the paper.Reasons for new version: The InSyDE package depends, in order to work, on determining the S-function. The problem is that, for very complicated 2ODEs, this can be very costly (computationally) or even unrealizable (within time and memory limits). We have developed a computationally cheaper way of determining the S-function that, in addition to being very efficient, is also surprisingly broad.Summary of revisions: We have, based on important new theoretical results presented on Appendix A: Supplementary Material, modified and improved the command Sfunction.

Soliton solutions and a bi-Hamiltonian structure of the fifth-order nonlocal reverse-spacetime Sasa-Satsuma-type hierarchy via the Riemann-Hilbert approach

<p>Our objective is to explore the intricacies of a nonlinear nonlocal fifth-order scalar Sasa-Satsuma equation in reverse spacetime which is rooted in a nonlocal $ 5 \times 5 $ matrix AKNS spectral problem. Starting with this spectral problem, we derive both local and nonlocal symmetry relations through rotations within a defined group. We then formulate a specific type of Riemann-Hilbert problem, facilitating the generation of soliton solutions. These solutions are generated by utilizing vectors that reside in the kernel of the matrix Jost solutions. Under the condition where reflection coefficients are null, the jump matrix reduces to the identity, leading to soliton solutions via the corresponding Riemann-Hilbert problem. The explicit formulas of these soliton solutions enable a comprehensive exploration of their dynamics.</p>

Light bullets in a nonlocal Rydberg medium with PT-symmetric moiré optical lattices

Realizing stable light bullets in high dimensions is a long-standing goal in the study of nonlinear optics. Here, we propose a scheme for the creation of stable light bullets (LBs) in a cold Rydberg atomic gas system with PT symmetry moiré optical lattices. Depending on the synergetic local and nonlocal Kerr nonlinearities, and PT symmetry moiré optical lattices, we obtain an appropriate nonlocal Rydberg long-range interaction and PT symmetry potential which can compensate for the diffraction and dispersion of the probe field. Numerical results show that different spatial and temporal distributions of LBs, including fundamental, multi-pole solitons, and vortical ones are uncovered, and their stabilities are evaluated through linear-stability analysis and direct simulation with perturbation. Furthermore, the solitons exhibit a quasi-elastic collision in the transversal direction during propagation. Our study provides a new route for manipulating LBs via controlled optical nonlinearities and PT symmetry moiré optical lattices in cold Rydberg gases.

Finding nonlocal Lie symmetries algorithmically

Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and nonlocal symmetries) admitted by a rational 2ODE if it presents a Liouvillian first integral. The procedure is based on an idea arising from the formal equivalence between the total derivative operator and the vector field associated with the 2ODE over its solutions (Cartan vector field). Basically, from the formal representation of a Lie symmetry it is possible to extract information that allows to use this symmetry practically (in the 2ODE integration process) even in cases where the formal operation cannot be performed, i.e., in cases where the symmetry is nonlocal. Furthermore, when the 2ODE in question depends on parameters, the procedure allows an analysis that determines the regions of the parameter space in which the integrable cases are located. We finish by applying the method to the 2ODE that represents a forced modified Duffing–Van der Pol oscillator (which presents chaos for arbitrary values of the parameters) and determining a region of the parameter space that leads to integrable cases, some not previously known.

Nonlocal symmetries, nonlocally related systems, similarity solutions and conservation laws of the Tzitzéica-Dodd-Bullogh equation

In this paper, a methodical procedure is utilized for the identification of nonlocal symmetries of the (1+1)-dimensional Tzitzéica-Dodd-Bullogh equation. Firstly, by introducing a set of canonical coordinates corresponding to the local Lie point symmetries, the considered partial differential eq. (PDE) is mapped to an invertibly equivalent PDE system. Furthermore, nonlocal symmetries are obtained from the inverse potential system of the invertibly equivalent PDE system. The exact solutions for the aforementioned PDE are acquired with the help of the extended generalized Kudryashov method corresponding to the admitted symmetries. In addition, the derivation of local conservation laws for the Tzitzéica-Dodd-Bullogh equation is obtained through the multiplier method. Moreover, using a symmetry-based technique and local conservation principles, a complete tree of nonlocally related PDE systems has been constructed.

Nonlocal symmetries and solutions of the (2+1) dimension integrable Burgers equation

The conservation laws of low-dimensional partial differential equations have been employed by Lou et al. to construct various high-dimensional equations, particularly focusing on high-dimensional integrable equations. However, solving these high-dimensional equations poses significant challenges. In this paper, the (2+1)-dimensional Burgers equation is studied by means of nonlocal symmetry method for the first time. For this high-dimensional equation, we establish a link with the exact solution of the (1+1)-dimensional Burgers equation through nonlocal symmetry. Furthermore, we successfully construct multiple exact solutions for the high-dimensional Burgers equation by leveraging the exact solution of the low-dimensional counterpart. We also give the corresponding images of several solutions to study the dynamic behavior of the equations.

On the investigation of a one-dimensional blood flow model in elastic arteries under symmetry analysis

In this paper, we consider a one-dimensional model of blood flow along the compliant arteries. With the help of the invariant function, we construct and classify the optimal system of subalgebras. Next, we reduced the given system of partial differential equations (PDEs) to the system of ordinary differential equations (ODEs) for each subalgebra and subsequently solved them exactly. Further, we investigate the evolutionary behavior of the average blood flow velocity and the cross-sectional area of the arteries; under the influence of the physical parameter [Formula: see text] graphically. Furthermore, we construct nonlocally related PDEs for the given system of PDEs, consisting of inverse potential systems (IPS) and potential systems. Finally, we classify the nonlocal symmetries arising from the potential system and IPS.