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Starting from the Darboux transformation (DT) related nonlocal symmetry of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation, the original symmetry is localized by introducing four field quantities. On the basis of the local symmetry, some novel group invariant solutions are derived by utilizing the classical symmetry reduction method. The result shows that the essential and unique role of the DT is to add an additional soliton to the fifth order nonlinear wave, which is the basic reduction of the (2+1)-dimensional CDGKS equation. As an illustration, the interactions between solitons and fifth order nonlinear waves expressed by Jacobi elliptic functions and incomplete elliptic integrals of the third kind are exhibited, and some of their dynamical properties are analyzed.

The nonlocal symmetry for the Whitham-Broer-Kaup (WBK) equation is obtained by the Painleve analysis. With the introduction of two auxiliary dependent variables, the nonlocal symmetry is localized to the Lie point symmetry. The reciprocal transformation is established by the localization nonlocal symmetry procedure and can be used to construct a link between the WBK equation and the usual Burgers equation. By using the similarity reductions related with the nonlocal symmetry of the Burgers equation, we derive the symmetry invariant solutions for the WBK equation. Furthermore, the power-series solutions are computed by using the power-series theory.

Starting from local coupled Hirota equations, we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem. The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair. By Lax pair, an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found. The solutions with specific properties are distinct from those of the local Hirota equation. In order to further describe the properties and the dynamic features of the solutions explicitly, several kinds of graphs are depicted.

In this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

An AB modified KdV (AB-mKdV) system which can be used to describe two-place event is studied in this manuscript. Because the AB-mKdV system is considered as a special reduction of the famous AKNS system, the properties of the AKNS system are first revealed by using symmetry analysis. The nonlocal symmetries related to truncated Painlevé expansion, the finite transformation, and the symmetry reduction solutions of the AKNS system are presented. The corresponding Bäcklund transformations and the interaction solutions of the AB-mKdV system are constructed based on the special reduction. The results demonstrate that the AB-mKdV system possesses many kinds of interaction solutions, such as the interactions between kink and soliton and kink and cnoidal waves. The soliton can be changed from bright to dark during propagation.

In this paper, the symmetry classification and symmetry reduction of a two-component reaction-diffusion system are investigated, the reaction-diffusion system can be reduced to system of ordinary differential equations, and the solutions and numerical simulation will be showed by examples.

Under investigation in this paper is a modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics/plasma physics. Painleve integrability is investigated. Soliton-cnoidal and resonant solutions are derived via the consistent tanh expansion method. The equation is reduced to certain symmetry reduction equations via the Lie point symmetry generators obtained though the Lie group method. Hyperbolic function and rational solutions are derived through those symmetry reduction equations.

Based on the modified direct method, the variable-coefficient perturbed mKdV equation is changed to the constant-coefficient perturbed mKdV equation. The truncated Painlevé method is applied to obtain the nonlocal symmetry of the constant-coefficient perturbed mKdV equation. By introducing one new dependent variable, the nonlocal symmetry can be localized to the Lie point symmetry. Thanks to the localization procedure, the finite symmetry transformation is presented by solving the initial value problem of the prolonged systems. Furthermore, many explicit interaction solutions among different types of solutions such as solitary waves, rational solutions, and Painlevé II solutions are obtained using the symmetry reduction method to the enlarged systems. Two special concrete soliton-cnoidal interaction solutions are studied in both analytical and graphical ways.

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.\udKeywords: liquid flow; hydrodynamic Euler equations; diffeomorphism group; Lie-Poisson structure; isentropic hydrodynamic invariants; vortex invariants; charged liquid fluid dynamics; symmetry reduction

The nonlocal symmetries are derived for the Korteweg–de Vries–negative-order Korteweg–de Vries equation from the Painlevé truncation method. The nonlocal symmetries are localized to the classical Lie point symmetries for the enlarged system by introducing new dependent variables. The corresponding similarity reduction equations are obtained with different constant selections. Many explicit solutions for the integrable equation can be presented from the similarity reduction.

Moving boundary problems of generalised Stefan type are considered for the Harry Dym equation via a Painleve II symmetry reduction. Exact solutions of such nonlinear boundary value problems are obtained in terms of Yablonski–Vorob’ev polynomials corresponding to an infinite sequence of values of the Painleve II parameter. The action of two kinds of reciprocal transformation on the moving boundary problems is described.

In this work we study one of the most important applications of symmetries to physical problems, namely the construction of conservation laws. Conservation laws have important place for applications of differential equations and solutions, also in all physics applications. And so, this study deals conservation laws of first- and second-type nonlinear (NL) reaction diffusion equations. We used Ibragimov’s approach for finding conservation laws for these equations. And then, we found exact solutions of first- and second-type NL reaction diffusion equations with Lie-point symmetries.

In the DFT community, it is common practice to use regular k -point grids (Monkhorst-Pack, MP) for Brillioun zone integration. Recently Wisesa et al. (2016) and Morgan et al. (2018) demonstrated that generalized regular (GR) grids offer an advantage over traditional MP grids. The difference is simple but effective. At the same k -point density, GR grids have greater symmetry and 60% fewer irreducible k-points. GR grids have not been widely adopted because one must search through a large number of candidate grids; in many cases, a brute force search could take hours. This work describes an algorithm that can quickly search over GR grids for those that have the most uniform distribution of points and the best symmetry reduction. The grids are ∼ 60% more efficient, on average, than MP grids and can now be generated on the fly in seconds.

In this article we consider a general system of complex short pulse equation (CSPE) that under certain nonlocal symmetry reduction yields a reverse space-time nonlocal complex short pulse equation (NL-CSPE). We apply matrix Darboux transformation to the associated Lax pair and construct multi-soliton solutions. K-soliton solution is expressed in terms of quasideterminant formula which enable us to compute explicit expressions of symmetry broken and symmetry preserving one- and two-soliton solutions for NL-CSPE. In addition to these solutions, we also obtain one-bright and interaction of two-bright solitons for classical CSPE. All these investigations end up with the conclusion that both symmetry preserving and broken solutions exist for NL-CSPE.

The purpose of this paper is to present a novel group-theoretic method for constructing Backlund transformations between systems of differential equations. Our approach is based upon the definition of Backlund transformations as integrable extensions of exterior differential systems. The construction of these transformations is obtained using the general theory of symmetry reduction of differential systems. Our method is then applied to differential systems which are integrable by the method of Darboux, and a detailed understanding of the Backlund transformations for these systems is obtained.

In this paper, by applying the direct symmetry method, we obtain the symmetry reductions, group invariant solutions and some new exact solutions of the Bogoyavlenskii equation, which include hyperbolic function solutions, trigonometric function solutions and power series solutions. We also give the conservation laws of the Bogoyavlenskii equation.

We study the scalar complex modified Korteweg-de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.

In natural and social science, many events happened at different space-times may be closely correlated. Two events, A (Alice) and B (Bob) are defined as correlated if one event is determined by another, say, $B=\hat{f} A$ for suitable $\hat{f}$ operators. A nonlocal AB-KdV system with shifted-parity ($P_s$, parity with a shift), delayed time reversal ($T_d$, time reversal with a delay) symmetry where $B=\hat{P_s}\hat{T_d} A$ is constructed directly from the normal KdV equation to describe two-area physical event. The exact solutions of the AB-KdV system, including $P_s T_d$ invariant and $P_s T_d$ symmetric breaking solutions are shown by different methods. The $P_s T_d$ invariant solution show that the event happened at $A$ will happen also at $B$. These solutions, such as single soliton solutions, infinitely many singular soliton solutions, soliton-cnoidal wave interaction solutions, and symmetry reduction solutions etc., show the AB-KdV system possesses rich structures. Also, a special B\"{a}cklund transformation related to residual symmetry is presented via the localization of the residual symmetry to find interaction solutions between the solitons and other types of the AB-KdV system., Comment: 20 pages

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques., Comment: laTeX, 31 pages, 5 figures

The symmetry reduction of higher order Painlev\'e systems is formulated in terms of Dirac procedure. A set of canonical variables that admit Dirac reduction procedure is proposed for Hamiltonian structures governing the ${A^{(1)}_{2M}}$ and ${A^{(1)}_{2M-1}}$ Painlev\'e systems for $M=2,3,...$., Comment: to appear in Phys. Lett. A

By means of the generalized direct method, we investigate the (2+1)-dimensional dispersive long wave equations. A relationship is constructed between the new solutions and the old ones and we obtain the full symmetry group of the (2+1)-dimensional dispersive long wave equations, which includes the Lie point symmetry group S and the discrete groups D. Some new forms of solutions are obtained by selecting the form of the arbitrary functions, based on their relationship. We also find an infinite number of conservation laws of the (2+1)-dimensional dispersive long wave equations.

Two coupled nonlinear Schrodinger equations describing the transmission of subpicosecond pulses in strongly birefringent fibers with Raman scattering are studied by a symmetry reduction method and a perturbative technique. Two families of approximate vector solitary waves are obtained. Reduction of the Raman frequency shift in birefringent fibers is established.

We construct condensate states encoding the continuum spherically symmetric quantum geometry of an horizon in full quantum gravity, i.e. without any classical symmetry reduction, in the group field theory formalism. Tracing over the bulk degrees of freedom, we show how the resulting reduced density matrix manifestly exhibits an holographic behavior. We derive a complete orthonormal basis of eigenstates for the reduced density matrix of the horizon and use it to compute the horizon entanglement entropy. By imposing consistency with the horizon boundary conditions and semiclassical thermodynamical properties, we recover the Bekenstein--Hawking entropy formula for any value of the Immirzi parameter. Our analysis supports the equivalence between the von Neumann (entanglement) entropy interpretation and the Boltzmann (statistical) one., Comment: 6 pages; published version

We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Amp\`ere structure, and Burgers'-type vortices are a canonical class of solutions associated with this structure. The mapping of such solutions, which are characterised by a linear dependence of the third component of the velocity on the coordinate defining the axis of rotation, to solutions of the incompressible equations in two dimensions is also shown to be an example of a symmetry reduction The Monge-Amp\`ere structure for incompressible flow in two dimensions is shown to be hypersymplectic., Comment: 22 pages, version 2: misprints are corrected, some cosmetic textual changes and references are added

In this paper, the Wick-type stochastic MKdV equation is researched by using symmetry reduction. And some stochastic soliton-like solutions are given via Hermite transformation.

A simple symmetry reduction procedure is repeatedly used to obtain infinitely many symmetries and then the exact solutions of the Burgers equation. Some sets of exact solutions such as the rational solutions, rational-kink solutions and error function solutions are explicitly given. As a byproduct the recursion operators can be re-obtained at the same time.

A general asymmetric Nizhnik-Novikov-Veselov (ANNV) family with an arbitrary function of high order group invariants is proposed. It is proved that the general ANNV family possesses a common infinite dimensional Kac-Moody-Virasoro symmetry algebra. The Kac-Moody-Virasoro group invariant solutions and the Kac-Moody group invariant solutions of the ANNV family are also studied.- PACS: 02.30.Jr, 02.30.Ik, 05.45.Yv.

Relative concentrations of four selected isomers of La@C 82 are computed from the Gibbs energy derived from partition functions supplied with parameters from density functional theory calculations. An agreement with experiment can be reached for temperatures roughly from 1000 to 1300 K when the C 2v species is the major isomer followed by an isomer that undergoes C 3v /C s symmetry reduction while the intrinsically C s species comes as a still less populated third product. It is suggested that the C 3v isomer can be suppressed in the condensed phase by its higher reactivity.

The harmonic analysis of the symmetry breaking is used to study the interaction and inter-layer motion of double-wall carbon nanotubes shells. The tremendous total symmetry reduction with respect to the high symmetry of the isolated layers results in low interaction: its decreasing with the symmetry breaking rate explains the observed small friction.

Based on classical Lie group method, the group for a general class of variable coefficient nonlinear diffusion–convection equation in (1 + 1) dimensions is obtained. New symmetries are found. Seven models have been studied.

Tunneling two-level systems are ubiquitous in amorphous solids, and form a major source of noise in systems such as nano-mechanical oscillators, single electron transistors, and superconducting qubits. Occurance of defect tunneling despite their coupling to phonons is viewed as a hallmark of weak defect-phonon coupling. This is since strong coupling to phonons results in significant phonon dressing and suppresses tunneling in two-level tunneling defects effectively. Here we determine the dynamics of a crystalline tunnelling defect strongly coupled to phonons incorporating the full 3D geometry in our description. We find that inversion symmetric tunnelling is not dressed by phonons whereas other tunnelling pathways are dressed by phonons and, thus, are suppressed by strong defect-phonon coupling. We provide the linear acoustic and dielectric response functions for a crystalline tunnelling defect for strong defect-phonon coupling. This allows direct experimental determination of the defect-phonon coupling. The singling out of inversion-symmetric tunneling states in single tunneling defects is complementary to their dominance of the low energy excitations in strongly disordered solids as a result of inter-defect interactions for large defect concentrations. This suggests that inversion symmetric two-level systems play a unique role in the low energy properties of disordered solids., Comment: 18 pages, 4 figures; minor modification, final version

The technique of equivariant moving frames is incorporated into the classical symmetry reduction method of ordinary finite difference equations. Using the recurrence relations for the finite difference invariants, computations are performed symbolically without relying on the coordinate expressions of the canonical variables and the difference invariants.

A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be used to perform symmetry reduction. The method generalizes one presented in a recent publication for the case of ordinary difference equations. In turn, it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables.

The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand, to periodic systems on the other. Boundary conditions are introduced to reduce theories on an infinite lattice to those on semi-infinite, or finite ones.

A successive symmetry reduction of the C 6 F 6 − radical anion along the active Jahn–Teller and pseudo-Jahn–Teller modes has been considered with ab initio calculations. Stationary points of the potential energy surface have been located and their types and interrelations determined. The energy values for stationary structures have been adjusted by MP2 calculations. The structures of C 2v and D 2 symmetry forming a pseudo-rotational surface are the lowest in energy. The pseudo-rotation includes six minima (C 2v ) and six transition states (D 2 ). The corresponding barrier height has been estimated as 0.2 kcal/mol from MP2/6-31+G*//6-31+G* calculations.

Inspired by the extended tanh function method and the direct method of symmetry reduction, we present a Burgers equation-based constructive method for solving nonlinear evolution equations (NLEEs). This method allows us to generate the solutions of NLEEs by the solutions of the Burgers equation. Applying this method to the NLEEs, we can obtain soliton-like solutions, multi-soliton-like solutions, formal periodic solutions and rational solutions, etc. We take the Boiti–Leon–Pempinelli equation as an example to illustrate the algorithm.

The infrared photodissociation spectra of Ag n (C 6 H 6 ) m and Ag n (C 6 D 6 ) m have been recorded near 10 μm for n = 3–7. For Ag n (C 6 H 6 ) m , we observe a feature around 1032 cm −1 , which is assigned to the allowed e lu fundamental of benzene. In addition, features are observed near 986 cm −1 in Ag n (C 6 H 6 ) m and 938 cm −1 in Ag n (C 6 D 6 ) m . These bands are assigned to the a 1g CC breathing mode - a transition which, while forbiddenfor a gas-phase benzene, is dipole allowed for benzene whose symmetry is reduced (to C 6v or lower) by adsorption.

Superconductors with a chiral p-wave pairing are of great interest because they could support Majorana modes that could enable the development of topological quantum computing technologies that are robust against decoherence. Sr₂RuO₄ is widely believed to be a chiral p-wave superconductor. Yet, the mechanism by which superconductivity emerges in this, and indeed most other unconventional superconductors, remains unclear. Here we show that the local superconducting transition temperature in the vicinity of lattice dislocations in Sr₂RuO₄ can be up to twice that of its bulk. This is all the more surprising for the fact that disorder is known to easily quench superconductivity in this material. With the help of a phenomenological theory that takes into account the crystalline symmetry near a dislocation and the pairing symmetry of Sr₂RuO₄, we predict that a similar enhancement should emerge as a consequence of symmetry reduction in any superconductor with a two-component order parameter.

The high resolution spectrum of AsH3 3ν1 and 2ν1+ν3 stretching overtone dyad was recorded and analyzed. The major vibration‐rotation parameters of these overtones were obtained. The result indicates that these overtone vibrational states are close to the local mode limit and the rotational levels show the tendency to approach an asymmetric top.

This paper presents a new method of constructing, certain classes of solutions of a system of partial differential equations (PDEs) describing the non-stationary and isentropic flow for an ideal compressible fluid. A generalization of the symmetry reduction method to the case of partially-invariant solutions (PISs) has been formulated. We present a new algorithm for constructing PISs and discuss in detail the necessary conditions for the existence of non-reducible PISs. All these solutions have the defect structure and are computed from four-dimensional symmetric subalgebras. These theoretical considerations are illustrated by several examples. Finally, some new classes of invariant solutions obtained by the symmetry reduction method are included. These solutions represent central, conical, rational, spherical, cylindrical and non-scattering double waves.

The residual symmetry relating to the truncated Painleve expansion of the Kadomtsev—Petviashvili (KP) equation is nonlocal, which is localized in this paper by introducing multiple new dependent variables. By using the standard Lie group approach, new symmetry reduction solutions for the KP equation are obtained based on the general form of Lie point symmetry for the prolonged system. In this way, the interaction solutions between solitons and background waves are obtained, which are hard to find by other traditional methods.

We have studied the maximal symmetry group admitted by the non-linear polywave equation lu = F(u). In particular, we establish that the equation in question admits the conformal group C(1, n) if and only if F(u) = λeu, n + 1 = 2l or F(u) = λu(n+1+2l)/(n+1-2l), n + 1 ≠ 2l. Symmetry reduction for the biwave equation 2u = λu-3 is carried out and some exact solutions are obtained.

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-B\"acklund vector field. Such generalization proves to be effective and enables us to construct principally new Ans\"atze reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations $u_t=u_{xx}+F(u,u_x)$ with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions can not be constructed by means of the symmetry reduction procedure., Comment: 12 pages, latex, needs amssymb., to appear in the "Journal of Physics A: Mathematical and General" (1995)

Biochemical reactions may be viewed as discrete event processes characterized by a finite number of states and transitions. These processes may be modeled as finite state transition systems where state transitions represent individual reaction events. The time-evolution of the state occupancy probabilities of such systems is described by the master equation. Since these systems often involve a large number of interactions, it can be difficult to construct the master equation for a model describing a system, and since the resulting models can involve a huge number of states, solving the associated master equation can be difficult or impossible. Here, we describe a method for the specification, construction, and reduction of finite state transition system models of biochemical processes using the symmetry and invariant manifold reduction techniques. The method allows a user to specify transition rules using an intuitive graphical representation, and to automatically construct the transition matrix of a differential equation characterizing exactly the dynamics of a model, with a potentially significant reduction in dimension when compared to the full master equation of the model. The application of the method to a biological process is illustrated by models describing a hypothetical ion-channel at several levels of complexity.

We present a symmetry-reduced version of the dielectrically-consistent reference interaction site model (DRISM) equation and an adaptation of the Labik–Malijevský–Voňka hybrid algorithm for its numerical solution. This approach is used for the calculation of site–site correlation functions of water, acetone and a water–acetone mixture. Compared to the traditional Picard iteration of non-reduced DRISM theories, savings of more than 90% in computational time are obtained. The resulting site–site pair-correlation functions are in reasonable agreement with computer simulations.

The authors make a case study of how a standard class of Hamiltonians for the fractional quantum Hall effect (modelled in a spherical geometry) can be explicitly symmetry reduced with respect to the full rotation group, and perform finite-size calculations for small particle numbers.

The nonlocal symmetry of the generalized fifth order KdV equation (FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Backlund transformation is obtained through Lie's first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions.

We study the Painleve property of the (1+1)-dimensional equations arising from the symmetry reduction for the (2+1)-dimensional ones. Firstly, we derive the similarity reduction of the (2+1)-dimensional potential Calogero—Bogoyavlenskii—Schiff (CBS) equation and Konopelchenko—Dubrovsky (KD) equations with the optimal system of the admitted one-dimensional subalgebras. Secondly, by analyzing the reduced CBS, KD, and Burgers equations with Painleve test, respectively, we find both the Painleve integrability, and the number and location of resonance points are invariant, if the similarity variables include all of the independent variables.