Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods

In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann‐Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi‐Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.     

Associate symmetries: A novel procedure for finding contact symmetries

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.     

Solving a nonlinear pde that prices real options using utility based pricing methods

We provide group invariant solutions to two nonlinear differential equations associated with the valuing of real options with utility pricing theory. We achieve these through the use of the Lie theory of continuous groups, namely, the classical Lie point symmetries. These group invariant solutions, constructed through the use of the symmetries that also leave the boundary conditions invariant, are consistent with the results in the literature. Thus it may be shown that Lie symmetry algorithms underlie many ad hoc methods that are utilised to solve differential equations in finance.     

Zero‐coupon bond prices in the Vasicek and CIR models: Their computation as group‐invariant solutions

We compute prices of zero‐coupon bonds in the Vasicek and Cox–Ingersoll–Ross interest rate models as group‐invariant solutions. Firstly, we determine the symmetries of the valuation partial differential equation that are compatible with the terminal condition and then seek the desired solution among the invariant solutions arising from these symmetries. We also point to other possible studies on these models using the symmetries admitted by the valuation partial differential equations. Copyright © 2007 John Wiley & Sons, Ltd.     

Symmetries of compatibility conditions for systems of differential equations

When symmetries of differential equations are applied, various types of associated systems of equations appear. Compatibility conditions of the associated systems expressed in the form of differential equations inherit Lie symmetries of the initial equations. Invariant solutions to compatibility systems are known as orbits of partially invariant and generic solutions involved in the Lie group foliation of differential equations and so on. In some cases Bäcklund transformations and differential substitutions connecting quotient equations for compatibility conditions and initial systems naturally arise. Besides, Ovsiannikov's orbit method for finding partially invariant solutions is essentially based on such symmetries.     

Einstein–Weyl structures from hyper–Kähler metrics with conformal Killing vectors

We consider four (real or complex) dimensional hyper-Kähler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein–Weyl structure which admits a shear-free geodesics congruence for which the twist is a constant multiple of the divergence. In this case the Einstein–Weyl equations reduce down to a single second order PDE for one function. The Lax representation, Lie point symmetries, hidden symmetries and the recursion operator associated with this PDE are found, and some group invariant solutions are considered.     

On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler–Poincaré theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied initially then it will be satisfied for all times. We then show how to solve these constraint equations in various examples to obtain evolution equations from the conservation laws. We also discuss some fluid conservation laws in the Euler–Poincaré theory that do not arise from Noether symmetries, and explain the relationship between the conservation laws obtained here, and the Kelvin–Noether theorem given in Sect. 4 of Holm et al. (Adv. Math. 137:1–81, 1998).     

Conservation laws and symmetries of semilinear radial wave equations

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrödinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg-de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved Hs norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions.     

Linking-Preserving Perturbations of Symmetric Functionals

We consider paths of functionals starting with one which is invariant under the action of an arbitrary group of symmetries. We give conditions for the existence of an unbounded sequence of critical values of the non-symmetric functional at the end of the path in terms of the growth of the critical values of the symmetric one. We apply this to obtain a multiplicity result for a system of elliptic equations whose symmetries are perturbed by a linear term and a non-homogeneous boundary condition.     

Conservation Laws,Symmetries, and Line Soliton Solutions of Generalized KP and Boussinesq Equations with <Emphasis Type="Italic">p</Emphasis>-Power Nonlinearities in Two Dimensions

Nonlinear generalizations of integrable equations in one dimension, such as the Korteweg–de Vries and Boussinesq equations with p-power nonlinearities, arise in many physical applications and are interesting from the analytic standpoint because of their critical behavior. We study analogous nonlinear p-power generalizations of the integrable Kadomtsev–Petviashvili and Boussinesq equations in two dimensions. For all p ≠ 0, we present a Hamiltonian formulation of these two generalized equations. We derive all Lie symmetries including those that exist for special powers p ≠ 0. We use Noether’s theorem to obtain conservation laws arising from the variational Lie symmetries. Finally, we obtain explicit line soliton solutions for all powers p > 0 and discuss some of their properties.     

Nonlocal symmetries and formulae for generation of solutions for a class of diffusion-convection equations

New formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion-convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation. We study in what particular ways additional Lie symmetries of diffusion-convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations. The formulae derived are used for the construction of exact solutions.     

Bifurcations on hemispheres

Summary It is now well known that the number of parameters and symmetries of an equation affects the bifurcation structure of that equation. The bifurcation behavior of reaction-diffusion equations on certain domains with certain boundary conditions isnongeneric in the sense that the bifurcation of steady states in these equations is not what would be expected if one considered only the number of parameters in the equations and the type of symmetries of the equations. This point was made previously in work by Fujii, Mimura, and Nishiura [6] and Armbruster and Dangelmayr [1], who considered reaction-diffusion equations on an interval with Neumann boundary conditions.As was pointed out by Crawford et al. [5], the source of this nongenericity is that reaction-diffusion equations are invariant under translations and reflections of the domain and, depending on boundary conditions, may naturally and uniquely be extended to larger domains withlarger symmetry groups. These extra symmetries are the source of the nongenericity. In this paper we consider in detail the steady-state bifurcations of reaction-diffusion equations defined on the hemisphere with Neumann boundary conditions along the equator. Such equations have a naturalO(2)-symmetry but may be extended to the full sphere where the natural symmetry group isO(3). We also determine a large class of partial differential equations and domains where this kind of extension is possible for both Neumann and Dirichlet boundary conditions.     

Symmetries of Integrable Difference Equations on the Quad-Graph

This paper describes symmetries of all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.     

Symmetry algebras of linear differential equations

The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.Tomsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 3–12, July, 1992.     

Vector hyperbolic equations with higher symmetries

We list eleven vector hyperbolic equations that have third-order symmetries with respect to both characteristics. This list exhausts the equations with at least one symmetry of a divergence form. We integrate four equations in the list explicitly, bring one to a linear form, and bring four more to nonlinear ordinary nonautonomous systems. We find the Bäcklund transformations for six equations.     

Conditional Lie–Bäcklund Symmetries and Invariant Subspaces to Nonlinear Diffusion Equations with Convection and Source

The conditional Lie–Bäcklund symmetry method is used to study the invariant subspace of the nonlinear diffusion equations with convection and source terms. We obtain a complete list of canonical forms for such equations which admit higher order conditional Lie–Bäcklund symmetries and multidimensional invariant subspaces. The functionally generalized separable solutions to the resulting equations are constructed due to the corresponding symmetry reductions. For most of the cases, they are reduced to solving finite‐dimensional dynamical systems.