Type II hidden symmetries through weak symmetries for some wave equations

The Type II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie-point symmetry. In [Gandarias RML. Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations. J Math Anal Appl 2008;348:752–9] it was shown that the provenance of the Type II Lie point hidden symmetries found for differential equations can be explained by considering weak symmetries or conditional symmetries of the original PDE.In this paper we analyze the connection between one of the methods analyzed in [Abraham-Shrauner B, Govinder KS. Provenance of Type II hidden symmetries from nonlinear partial differential equations. J Nonlin Math Phys 2006;13:612–22] and the weak symmetries of some partial differential equations in order to determine the source of these hidden symmetries. We have considered some of the models presented in [Abraham-Shrauner B, Govinder KS. Provenance of Type II hidden symmetries from nonlinear partial differential equations. J Nonlin Math Phys 2006;13:612–22], as well as the linear two-dimensional and three-dimensional wave equations [Abraham-Shrauner B, Govinder KS, Arrigo JA. Type II hidden symmetries of the linear 2D and 3D wave equations. J h Phys A Math Theor 2006;39:5739–47].     

On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation are successfully obtained, which contain soliton-like and periodic-like solutions This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

ADM-Padé technique for the nonlinear lattice equations

ADM-Padé technique is a combination of Adomian decomposition method (ADM) and Padé approximants. We solve two nonlinear lattice equations using the technique which gives the approximate solution with higher accuracy and faster convergence rate than using ADM alone. Bell-shaped solitary solution of Belov–Chaltikian (BC) lattice and kink-shaped solitary solution of the nonlinear self-dual network equations (SDNEs) are presented. Comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the technique.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution 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.     

Recursion Operator of the Narita–Itoh–Bogoyavlensky Lattice

We construct a recursion operator for the family of Narita–Itoh–Bogoyavlensky infinite lattice equations using its Lax presentation and present their mastersymmetries and bi‐Hamiltonian structures. We show that this highly nonlocal recursion operator generates infinitely many local symmetries.     

Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

A straightforward algorithm for the symbolic computation of generalized (higher‐order) symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations. With our Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonlocal symmetries. Heuristic approach

A constructive method for constructing nonlocal symmetries of differential equations based on the Lie—Bäcklund theory of groups is developed. The concept of quasilocal symmetries is introduced. With the help of this method nonlocal symmetries of differential equations of the type of nonlinear thermal conductivity and gas dynamics are studied.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 34, pp. 3–83, 1989.     

Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.     

Polarized Hessian covariant: Contribution to pattern formation in the Föppl–von Kármán shell equations

We analyze the structure of the Föppl–von Kármán shell equations of linear elastic shell theory using surface geometry and classical invariant theory. This equation describes the buckling of a thin shell subjected to a compressive load. In particular, we analyze the role of polarized Hessian covariant, also known as second transvectant, in linear elastic shell theory and its connection to minimal surfaces. We show how the terms of the Föppl–von Kármán equations related to in-plane stretching can be linearized using the hodograph transform and relate this result to the integrability of the classical membrane equations. Finally, we study the effect of the nonlinear second transvectant term in the Föppl–von Kármán equations on the buckling configurations of cylinders.     

The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique

The purpose of this study is to implement Adomian–Pade (Modified Adomian–Pade) technique, which is a combination of Adomian decomposition method (Modified Adomian decomposition method) and Pade approximation, for solving linear and nonlinear systems of Volterra functional equations. The results obtained by using Adomian–Pade (Modified Adomian–Pade) technique, are compared to those obtained by using Adomian decomposition method (Modified Adomian decomposition method) alone. The numerical results, demonstrate that ADM–PADE (MADM–PADE) technique, gives the approximate solution with faster convergence rate and higher accuracy than using the standard ADM (MADM).     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

非齐次Toda晶格的对称,精确解和可积性

研究非齐次Toda晶格，即一类非齐次非线性微分差分方程的对称与可积性。给出了这一类方程的Lie点对称，条件对称和精确解。给出这类方程与Toda晶格之间的可逆点变换，从而表明这一类方程是可积的。     

Exp-function method for constructing exact solutions of Sharma–Tasso–Olver equation

In this paper we use the Exp-function method for the analytic treatment of Sharma–Tasso–Olver equation. New solitonary solutions are formally derived. Change of parameters, which drastically changes the characteristics of the equations, is examined. It is shown that the Exp-function method provides a powerful mathematical tool for solving high-dimensional nonlinear evolutions in mathematical physics. The proposed schemes are reliable and manageable.     

The compact and noncompact structures for two types of generalized Camassa–Holm–KP equations

The objective of this paper is to investigate two types of generalized nonlinear Camassa–Holm–KP equations in (2+1) dimensional space. Compactons, solitons, solitary patterns, periodic solutions and algebraic travelling wave solutions are expressed analytically under various circumstances. The conditions that cause the qualitative change in the physical structures of the solutions are emphasized.     

Master partial differential equations for a Type II hidden symmetry

An approach for determining a class of master partial differential equations from which Type II hidden point symmetries are inherited is presented. As an example a model nonlinear partial differential equation (PDE) reduced to a target PDE by a Lie symmetry gains a Lie point symmetry that is not inherited (hidden) from the original PDE. On the other hand this Type II hidden symmetry is inherited from one or more of the class of master PDEs. The class of master PDEs is determined by the hidden symmetry reverse method. The reverse method is extended to determine symmetries of the master PDEs that are not inherited. We indicate why such methods are necessary to determine the genesis of Type II symmetries of PDEs as opposed to those that arise in ordinary differential equations (ODEs).     

The integrable coupling system of a 3 × 3 discrete matrix spectral problem

Integrable coupling with six potentials is first proposed by coupling a given 3 × 3 discrete matrix spectral problem. It is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebras, the Hamiltonian form is deduced for the lattice equations in the resulting hierarchy. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian system.     

Applications of Feller–Reuter–Riley Transition Functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.     

Modified ‐expansion method for finding exact wave solutions of the coupled Klein–Gordon–Schrödinger equation

The coupled Klein–Gordon–Schrödinger equation is reduced to a nonlinear ordinary differential equation (ODE) by using Lie classical symmetries, and various solutions of the nonlinear ODE are obtained by the modified ‐expansion method proposed recently. With the aid of solutions of the nonlinear ODE, more explicit traveling wave solutions of the coupled Klein–Gordon–Schrödinger equation are found out. The traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. Copyright © 2012 John Wiley & Sons, Ltd.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.