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In this work, we investigate a symmetry reduction of the recently discovered (3 + 1)-dimensional equation of the Monge-Ampère type. This equation forms a bi-Hamiltonian system using Magri's theorem when expressed in the two-component form. We select a particular linear combination of the Lie point symmetries belonging to this system to conduct symmetry reduction, resulting in a new (2 + 1)-dimensional system in two-component form. Lagrangian and first Hamiltonian densities are then calculated. We employ Dirac's theory of constraints to obtain symplectic and first Hamiltonian operators. Subsequently, we transform the symmetry condition of the reduced system into a skew-factorized form to determine the recursion operator. Applying the recursion operator to the first Hamiltonian operator yields the second Hamiltonian operator. We demonstrate that the reduced system is a bi-Hamiltonian integrable system in the sense of Magri. Lie point symmetries of the reduced system are identified. Finally, we calculate integrals of motion using the inverse Noether theorem and prove that they have the total divergence form. [ABSTRACT FROM AUTHOR]

In this study, we explore the effect of integrated auxiliary rods at varying angles to the primary cavity rod on the dispersion characteristics of the photonic crystal coupled cavity waveguide (PC CCW). Here, it is intended to break the symmetry of the cavity region by introducing auxiliary rods which gives the degree of freedom for tuning effective index of the PC waveguide. Furthermore, rotational angle variations of auxiliary rods reveal slow light operation of the PC CCW where the group index is maximized and group velocity dispersion, as well as the third-order dispersion, are minimized. In addition, auxiliary rods with a broken symmetry increase not only group index but also operating bandwidth and accordingly increase group bandwidth product by 675%. Leveraging these results, we demonstrate effective rainbow trapping by manipulating the auxiliary rod angles in photonic crystal coupled cavity waveguides. Our results have encouraging implications for optical buffering, multiplexing, demultiplexing, advanced time-domain and spatial signal processing. [ABSTRACT FROM AUTHOR]

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph K m , n , extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr (K 10 , n) ≥ 4.87057 n 2 - 10 n , cr (K 11 , n) ≥ 5.99939 n 2 - 12.5 n , cr (K 12 , n) ≥ 7.25579 n 2 - 15 n , cr (K 13 , n) ≥ 8.65675 n 2 - 18 n for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite. [ABSTRACT FROM AUTHOR]

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions 11 through 23. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem. [ABSTRACT FROM AUTHOR]

We perform the procedure of symmetry reduction of (1+3)-dimensional inhomogeneous Monge–Ampère equation to algebraic equations. Some results obtained with the use of the classification of three-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1, 4) are presented. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Riemann–Liouville (R-L) time-fractional (2 + 1) dimensional extended quantum Zakharov–Kuznetsov (EQZK) equation by using the Lie symmetry method which arises in hydrodynamic that describes the nonlinear propagation of the quantum ion-acoustic waves. By using its symmetry, we convert the equation under consideration to a fractional order non-linear ordinary differential equation (ODE). In this reduced ODE, we use a special type of derivative which is known as Erdélyi–Kober (EK) derivative. This enables us to obtain explicit solutions with convergence analysis of the considered problem. By using Ibragimov's conservation laws theorem, we compute the conservation laws of the problem under investigation. Moreover, by employing the two potent methods explicit power series and ( 1 G ′ )-expansion technique, we get the explicit solutions to the problem under discussion. This analysis leads to the derivation of various key findings, including the identification of symmetries, the establishment of similarity reductions involving the EK fractional differential operator, the determination of exact solutions, and the formulation of conservation laws for the considered equation. We have confidence that these remarkable findings can provide valuable insights and contribute to the exploration of additional evolutionary mechanisms associated with the studied equation. [ABSTRACT FROM AUTHOR]

A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed. [ABSTRACT FROM AUTHOR]

We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g. The key idea advocated for in this article is that the "correct" notion of positivity on a *-algebra A is not necessarily the algebraic one, for which positive elements are sums of Hermitian squares a*a with a∈A, but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra Ared should be such that states on Ared are obtained as reductions of certain states on A. We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold M, reduction of the Weyl algebra with respect to translation symmetry, and reduction of the polynomial algebra with respect to a U(1)-action. [ABSTRACT FROM AUTHOR]

We address the problem of symmetry reduction of optimal control problems under the action of a finite group from a measure relaxation viewpoint. We propose a method based on the moment-Sum of Squares (SOS) aka Lasserre hierarchy which allows one to significantly reduce the computation time and memory requirements compared to the case without symmetry reduction. We show that the recovery of optimal trajectories boils down to solving a symmetric parametric polynomial system. Then we illustrate our method on the symmetric integrator and the time-optimal inversion of qubits. [ABSTRACT FROM AUTHOR]

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be y or z independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given. [ABSTRACT FROM AUTHOR]

We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the A n , B n , and D n case where we use so-called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, to deduce that the description of the cones of sums of squares of fixed degree 2 d stabilizes with n > 2 d. Furthermore, in cases of small degree, we are able to analyze these cones more explicitly and compare them to the cones of non-negative forms. [ABSTRACT FROM AUTHOR]

We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on $ SE(2) $ in the presence of a static obstacle.

In this paper, a modified Korteweg–de Vries equation with a quartic nonlinear term in two-electron temperature plasmas is studied. Firstly, the symmetries are constructed using the generalized symmetry method. In addition, the potential equation is studied, it can been found that this equation does not posses potential symmetries. Meanwhile, the symmetry reduction and group-invariant solutions are derived rely on the one-dimensional optimal system of subalgebras. Finally, conservation laws are showed resort by the multipliers method, three polynomial type conserved quantities are presented, also the Hamiltonian form is obtained. In particular, reciprocal Bäcklund transformations for conservation laws are obtained for the first time.

An extended (2+1)-dimensional shallow water wave (SWW) model governs the evolution of nonlinear shallow water wave propagation in two spatial and a temporal coordinate. The multi-linear variable separation approach is applied to the SWW equation. The variable separation solution consisting of two arbitrary functions is given. By choosing the arbitrary functions as the exponential and trigonometric forms, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions and periodic waves are graphically and analytically studied. The results enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

This work presents a generalized hyper-elastic rod wave (gHRW) equation from the Lie symmetry method’s standpoint. The equation illustrates dispersive waves generating in hyper-elastic rods. Using multiplier approach we find conserved vectors of the underlying equation. We subsequently obtain first integrals of the conserved vectors under the time–space group invariant u(t,x)=H(x−νt). Finally, by analysing various attainable instances of the arbitrary coefficient function g(u), we perform symmetry reductions of gHRW equation to lower order ordinary differential equations and in some instances obtain analytic solutions for special values of arbitrary constants.

In this paper we construct a hierarchy of multivariate polynomial approximation kernels for uniformly continuous functions on the hypercube via semidefinite programming. We give details on the implementation of the semidefinite programs defining the kernels. Finally, we show how symmetry reduction may be performed to increase numerical tractability. [ABSTRACT FROM AUTHOR]

We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on S E (2) in the presence of a static obstacle. [ABSTRACT FROM AUTHOR]

We study symmetry reductions of nonlinear partial differential equations that can be used for describing diffusion processes in heterogeneous medium. We find ansatzes reducing these equations to systems of ordinary differential equations. The ansatzes are constructed using generalized symmetries of second-order ordinary differential equations. The method applied gives the possibility to find exact solutions which cannot be obtained by virtue of the classical Lie method. Such solutions are constructed for nonlinear diffusion equations that are invariant with respect to one-parameter and two-parameter Lie groups of point transformations. We prove a theorem relating the property of invariance of a found solution to the dimension of the Lie algebra admitted by the corresponding equation. We also show that the method is applicable to non-evolutionary partial differential equations and ordinary differential equations.

We study the reduction by symmetries for optimality conditions in optimal control problems of left-invariant affine control systems with partial symmetry breaking cost functions. We recast the optimal control problem as a constrained problem with a partial symmetry breaking Hamiltonian and we obtain the reduced optimality conditions for normal extrema from Pontryagin's Maximum Principle and a Lie--Poisson bracket on the reduced state space. We apply the results to an energy-minimum obstacle avoidance problems. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study soliton solutions for nonlocal Kundu-nonlinear Schrödinger (Kundu-NLS) equation via the Darboux transformation. The nonlocal Kundu-NLS equation can be obtained through a symmetry reduction r (x , t) = q ∗ (- x , t) . The form of N-soliton solutions for the nonlocal Kundu-NLS equation can be investigated via the one-fold and n-fold Darboux transformation. Particularly, from the Darboux transformation of the nonlocal Kundu-NLS equation, we obtain some exact solutions for the nonlocal Kundu-NLS equation with different spectral parameters and corresponding graphs are given. [ABSTRACT FROM AUTHOR]

Design of pyroelectric crystals decoupled from piezoelectricity is not only a topic of scientific curiosity but also demonstrates effects in principle that have the potential to be technologically advantageous. Here we report a new method for the design of such materials. Thus, the co‐doping of centrosymmetric crystals with tailor‐made guest molecules, as illustrated by the doping of α‐glycine with different amino acids (Threonine, Alanine and Serine). The polarization of those crystals displays two distinct contributions, one arising from the difference in dipole moments between guest and host and the other from the displacement of host molecules from their symmetry‐related positions. These contributions exhibit different temperature dependences and response to mechanical deformation. Thus, providing a proof of concept for the ability to design pyroelectric materials with reduced piezoelectric coefficient (d22) to a minimal value, below the resolution limit of the method (<0.005 pm/V). [ABSTRACT FROM AUTHOR]

A common computational approach for polynomial optimization problems (POPs) is to use (hierarchies of) semidefinite programming (SDP) relaxations. When the variables in the POP are required to be nonnegative – as is the case for combinatorial optimization problems, for example – these SDP problems typically involve nonnegative matrices, i.e. they are conic optimization problems over the doubly nonnegative cone. The Jordan reduction, a symmetry reduction method for conic optimization, was recently introduced for symmetric cones by Parrilo and Permenter [Mathematical Programming 181(1), 2020]. We extend this method to the doubly nonnegative cone, and investigate its application to known relaxations of the quadratic assignment and maximum stable set problems. We also introduce new Julia software where the symmetry reduction is implemented. [ABSTRACT FROM AUTHOR]

In this paper, we study a fourth-order nonlinear symmetric regularized long-wave equation which arises in several physical applications including ion sound waves in plasma. This model also describes nonlinear ion-acoustic and space-charge waves. Lie group analysis and the tanh-coth method is utilized for the construction of solitary wave solutions to the equation. Based on each symmetry, we reduce the considered partial differential equation into different ordinary differential equations. The obtained equations are solved and new solitary wave patterns are reported. To understand the physical interpretation, some solitary wave structures are exhibited graphically. Also, by using the multiplier approach, we constructed the local conservation laws of the considered equation.

We study the relationship between structural properties of the two-dimensional nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P(1,4) and the properties of reduced equations for the (1+3)-dimensional homogeneous Monge-Ampère equation. In this paper, we present some of the results obtained concerning symmetry reduction of the equation under investigation to identities. Some classes of the invariant solutions (with arbitrary smooth functions) are presented.

The flow of a viscous incompressible fluid outflowing from a uniformly moving point source is considered. An exact solution to the problem is found in the way that the velocity decreases inversely with the radial coordinate. It is shown that a spherical volume of fluid is carried away by the source, the radius of which is inversely proportional with respect to the velocity of motion. In this case, a cylindrical discontinuity arises in the region of forming a wake behind the body, the dimensions of which are determined by the magnitude of the external pressure and do not depend on the velocity of the source. The obtained solutions are governed by hydrodynamical fields of flows which can be recognized as special invariants at symmetry reduction. [ABSTRACT FROM AUTHOR]

We give a non-commutative Positivstellensatz for C P n : The (commutative) ⁎-algebra of polynomials on the real algebraic set C P n with the pointwise product can be realized by phase space reduction as the U (1) -invariant polynomials on C 1 + n , restricted to the real (2 n + 1) -sphere inside C 1 + n , and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U (1) -invariant polynomials on C 1 + n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative ⁎-algebra of polynomials on C 1 + n , the Weyl algebra, and give an algebraic description of the real-valued U (1) -invariant polynomials that are positive in certain ⁎-representations on Hilbert spaces of holomorphic sections of line bundles over C P n. It is especially noteworthy that the non-commutative result applies not only to strictly positive, but to all positive (semidefinite) elements. As an application, all ⁎-representations of the quantization of the polynomial ⁎-algebra on C P n , obtained e.g. through phase space reduction or Berezin–Toeplitz quantization, are determined. [ABSTRACT FROM AUTHOR]

In this paper, the Sharma-Tasso-Olver-Burgers (STOB) system is analyzed by the Lie point symmetry method. The hypergeometric wave solution of the STOB equation is derived by symmetry reductions. In the meantime, the consistent tanh expansion (CTE) method is applied to the STOB equation. An nonauto-Bäcklund (BT) theorem that includes the over-determined equations and the consistent condition is obtained by the CTE method. By using the nonauto-BT theorem, the interactions between one-soliton and the cnoidal wave, and between one-soliton and the multiple resonant soliton solutions, are constructed. The dynamics of these novel interaction solutions are shown both in analytical and graphical forms. The results are potentially useful for explaining ocean phenomena. [ABSTRACT FROM AUTHOR]

In this article, the integrable nonlocal nonlinear variable coefficient Newell–Whitehead (NW) equation is investigated for the first time. First, the variable coefficient nonlocal NW equation is constructed with the aid of symmetry reduction and Lax pair. On this basis, the Darboux transformation of the variable coefficient nonlocal NW equation is studied. Then, some exact solutions are obtained by applying the Darboux transformation. The results show that the variable coefficient equation has more general solutions than its constant coefficient form. Finally, the solutions of the variable coefficient nonlocal NW equation are given when the coefficient function takes on special values, and the structural features of the solutions are visualized in images. [ABSTRACT FROM AUTHOR]

In present paper, Lie group analysis is applied to consider vector field and symmetry reduction on forth-order time fractional KdV equation, power series solution and the convergence are investigated. Stability analysis of trivial solution to the reduction differential equation is showed by constructing appropriate Lyapunov function. Conservation laws of the equation are well constructed with a detailed derivation making use of Noether's operator. [ABSTRACT FROM AUTHOR]

Exact solutions for the shallow water equations in two spatial dimensions are obtained from a matrix formulation of the governing system. These fully nonlinear long waves exhibit linear velocity and quadratic free surface displacement fields in the spatial variables. No breaking phenomenon is observed. A special moving boundary can be defined where a fixed mass of fluid inside this circular region may display a rogue wave type behavior. The fluid first converges towards the center of a domain, and the amplitude of the free surface will attain a finite maximum. Subsequently, the fluid reverses paths and rushes away from the center. The kinetic and potential energy of the rogue wave can be computed analytically for all time. As the shallow water equations are employed, these rogue waves are fully nonlinear. There is no restriction on the amplitude, quite unlike the weakly nonlinear assumptions of the nonlinear Schrödinger and Korteweg—de Vries theories. Furthermore, the maximum is truly localized in two spatial dimensions horizontally and the wave is not a ‘long-crested’ one.

Reductions make it possible to reduce the solution of a PDE to solving an ODE. The best known are the traveling wave, self-similar and symmetry reductions. Classical and non-classical symmetries are also used to construct reductions, as is the Clarkson–Kruskal direct method. Recently, authors have proposed a method for constructing reductions of PDEs with two independent variables based on the idea of invariance. The proposed method in this work is a modification of the Clarkson–Kruskal direct method and expands the possibilities for its application. The main result of this article consists of a method for constructing reductions that generalizes the previously proposed approach to the case of three independent variables. The proposed method is used to construct reductions of the unsteady axisymmetric boundary layer equation to ODEs and simpler PDEs. All reductions of this equation were obtained. [ABSTRACT FROM AUTHOR]

This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds. [ABSTRACT FROM AUTHOR]

We introduce the notion of a nonlinear splitting on a fibre bundle as a generalization of an Ehresmann connection. We present its basic properties and we pay attention to the special cases of affine, homogeneous and principal nonlinear splittings. We explain where nonlinear splittings appear in the context of Lagrangian systems and Finsler geometry and we show their relation to Routh symmetry reduction, submersive second-order differential equations and unreduction. We define a curvature map for a nonlinear splitting, and we indicate where this concept appears in the context of nonholonomic systems with affine constraints and Lagrangian systems of magnetic type. [ABSTRACT FROM AUTHOR]

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from ℂ 1 + n with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space ℂ ℙ n and the complex hyperbolic disc n . We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of "polynomial" functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on ℂ ℙ n and n . More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the ∗ -involution of pointwise complex conjugation. [ABSTRACT FROM AUTHOR]

The nonlocal symmetry of the new integrable (2 + 1) -dimensional Boussinesq equation is studied by the standard truncated Painlevé expansion. This nonlocal symmetry can be localized to the Lie point symmetry of the prolonged system by introducing two auxiliary dependent variables. The corresponding finite symmetry transformation and similarity reduction related to the nonlocal symmetry of the new integrable (2 + 1) -dimensional Boussinesq equation are studied. The rational solution, the triangle solution, two solitoff-interaction solution and the soliton–cnoidal interaction solutions for the new (2 + 1) -dimensional Boussinesq equation are presented analytically and graphically by selecting the proper arbitrary constants. [ABSTRACT FROM AUTHOR]

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on SL(2, ℝ). Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups to identify exactly automorphic structures on which our results apply. [ABSTRACT FROM AUTHOR]

This paper analyzes the constraint structure of a class of degenerate second-order particle Lagrangian that includes chiral oscillator, noncommutative oscillator, and two examples from reduced topologically massive gravity. For even-dimensional configuration spaces with maximal nondegeneracy, Dirac bracket is defined solely by coefficient field of highest derivative whereas for odd dimensions almost all fields may contribute. Ostrogradskii's theorem on energy instability is discussed. Results of Dirac analysis are used to identify ghost degrees of freedom. Translational symmetries are used to construct first-order variational formalisms for oscillator examples, thereby making them ghost-free. [ABSTRACT FROM AUTHOR]