Self-adjoint Equations Research Articles

On the nonlinear self-adjointness of the Zakharov–Kuznetsov equation

A new general theorem, which does not require the existence of Lagrangians, allows to compute conservation laws for an arbitrary differential equation. This theorem is based on the concept of self-adjoint equations for nonlinear equations. In this paper we show that the Zakharov–Kuznetsov equation is self-adjoint and nonlinearly self-adjoint. This property is used to compute conservation laws corresponding to the symmetries of the equation. In particular the property of the Zakharov–Kuznetsov equation to be self-adjoint and nonlinearly self-adjoint allows us to get more conservation laws.

Optimal control of self-adjoint nonlinear operator equations in Hilbert spaces

In this paper, we formulate and study a general optimal control problem governed by nonlinear operator equations described by unbounded self-adjoint operators in Hilbert spaces. This problem extends various particular control models studied in the literature, while it has not been considered before in such a generality. We develop an efficient way to construct a finite-dimensional subspace extension of the given self-adjoint operator that allows us to design the corresponding adjoint system and finally derive an appropriate counterpart of the Pontryagin Maximum Principle for the constrained optimal control problem under consideration by using the obtained increment formula for the cost functional and needle type variations of optimal controls.

Lagrangians for self-adjoint and non-self-adjoint equations

It is relatively easy to obtain a variational formulation for a self-adjoint equation, while it is difficult or impossible to do this for a non-self-adjoint one. This work suggests an alternative approach to the derivation of the Lagrangian for both cases. A trial-Lagrangian is constructed, and the exact variational formulation can be obtained after the identification of the unknown function involved in the trial-Lagrangian.

New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order

In a recent communication Ibragimov introduced the concept of nonlinearly self-adjoint differential equation [Ibragimov NH. Nonlinear self-adjointness and conservation laws. J Phys A Math Theor 2011;44:432002 (8pp.)]. In this paper a nonlinear self-adjoint classification of a general class of fifth-order evolution equation with time dependent coefficients is presented. As a result five subclasses of nonlinearly self-adjoint equations of fifth-order and four subclasses of nonlinearly self-adjoint equations of third-order are obtained. From the Ibragimov’s theorem on conservation laws [Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333:311–28] conservation laws for some of these equations are established.

Some conservation laws for a forced KdV equation

The Korteweg–de Vries equation with a forcing term is established by recent studies as a simple mathematical model by which the physics of a shallow layer of fluid subject to external forcing is described. It serves as an analytical model of tsunami generation by sub-marine landslides. In this paper, we derive the classical Lie symmetries admitted by the forced KdV equation. Looking for travelling wave solutions we find that the forced KdV equation has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions. Hence the Korteweg–de Vries equation with a forcing term has plenty of periodic waves and solitary waves. These solutions are derived from the solutions of a simple nonlinear ordinary differential equation. The first author has introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by Ibragimov in 2006. In a previous paper we found a class of weak self-adjoint forced Korteweg–de Vries equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved by Ibragimov, the Lie symmetries and the new concept of weak self-adjointness we find some conservation laws for some of these partial differential equations.

Nonlinear self-adjointness of a generalized fifth-order KdV equation

The new concepts of self-adjoint equations formulated by Ibragimov and Gandarias are applied to a class of fifth-order evolution equations. Then, from Ibragimov’s theorem on conservation laws, conservation laws for the generalized Kawahara equation, simplified Kahawara equation and modified simplified Kawahara equation are established.

Weak self-adjointness and conservation laws for a porous medium equation

The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006, 2007) [4,7]. In Ibragimov (2007) [6] a general theorem on conservation laws was proved. In Gandarias (2011) [3] we generalized the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. In this paper we find the subclasses of weak self-adjoint porous medium equations. By using the property of weak self-adjointness we construct some conservation laws associated with symmetries of the differential equation.

Some weak self-adjoint Hamilton–Jacobi–Bellman equations arising in financial mathematics

In Gandarias (2011) [12] one of the present authors has introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by Ibragimov (2006) [11]. In this paper we find a class of weak self-adjoint Hamilton–Jacobi–Bellman equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved in Ibragimov (2007) [9] and the new concept of weak self-adjointness (Gandarias, 2011) [12] we find conservation laws for some of these partial differential equations.

Self-adjoint sub-classes of third and fourth-order evolution equations

In this work a class of self-adjoint quasilinear third-order evolution equations is determined. Some conservation laws of them are established and a generalization on a self-adjoint class of fourth-order evolution equations is presented.

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.

Some abstract critical point theorems for self-adjoint operator equations and applications

By using the index theory for linear bounded self-adjoint operators in a Hilbert space related to a fixed self-adjoint operator A with compact resolvent, the authors discuss the existence and multiplicity of solutions for (nonlinear) operator equations, and give some applications to some boundary value problems of first order Hamiltonian systems and second order Hamiltonian systems.

A method for computing acceleration-dependent Lagrangians

The problem of constructing an acceleration-dependent Lagrangian from the equations of motion is studied. A method for the computation of an acceleration-dependent Lagrangian from the self-adjoint equations is presented. Two examples are given to illustrate the application of the results.

Banded Matrices and Discrete Sturm-Liouville Eigenvalue Problems

We consider eigenvalue problems for self-adjoint Sturm-Liouville difference equations of any even order. It is well known that such problems with Dirichlet boundary conditions can be transformed into an algebraic eigenvalue problem for a banded, real-symmetric matrix, and vice versa. In this article it is shown that such a transform exists for general separated, self-adjoint boundary conditions also. But the main result is an explicit procedure (algorithm) for the numerical computation of this banded, real-symmetric matrix. This construction can be used for numerical purposes, since in the recent paper by Kratz and Tentler (2008) there is given a stable and superfast algorithm to compute the eigenvalues of banded, real-symmetric matrices. Hence, the Sturm-Liouville problems considered here may now be treated by this algorithm.

On the stability of the σ-scheme with transparent boundary conditions for parabolic equations

Initial-boundary value problems for self-adjoint parabolic equations on a semiaxis and a semibounded strip are considered. For finite-difference σ-schemes, an alternative method for stating approximate transparent boundary conditions is suggested and conditions ensuring unconditional stability in the energy norm with respect to the initial data and free terms for a weight σ ≥ 1/2 are presented. The validity of these stability conditions in the case of discrete transparent boundary conditions is proved (by several methods), and the derivation of the latter conditions is revisited.

Q-Dominant and q-recessive matrix solutions for linear quantum systems

In this study, linear second-order matrix q-difference equations are shown to be formally self-adjoint equations with respect to a certain inner prod- uct and the associated self-adjoint boundary conditions. A generalized Wronskian is introduced and a Lagrange identity and Abel's formula are established. Two reduction-of-order theorems are given. The analysis and characterization of q- dominant and q-recessive solutions at infinity are presented, emphasizing the case when the quantum system is disconjugate.

Oscillation Criteria for Second-order Nonlinear Self-adjoint Differential Equations

Our concern is to solve the oscillation problem for the nonlinear self-adjoint equation (a(t)x')' + b(t)g(x) = 0, where g(x) satisfies the Signum condition xg(x) > 0 if x 6= 0, but is not imposed such monotonicity as superlinear or sublinear. The problem has not been solved for the critical cases: liminf |x|.0 g(x) x < 1 4 < limsup |x|.0 g(x) x , and liminf |x|.8 g(x) x < 1 4 < limsup |x|.8 g(x) x , which are more difficult, by now. We concentrate our attention on this point and give some answers. Sufficient conditions are given for all nontrivial solutions to be oscillatory

Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions

This paper is concerned with periodic and antiperiodic boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues of these two different boundary value problems is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. In addition, a representation of solutions of a nonhomogeneous linear equation with initial conditions is given.

Adjoint pairs of differential-algebraic equations and Hamiltonian systems

We consider linear homogeneous differential-algebraic equations A ( D x ) ′ + B x = 0 and their adjoints − D ∗ ( A ∗ x ) ′ + B ∗ x = 0 with well-matched leading coefficients in parallel. Assuming that the equations are tractable with index less than or equal to 2, we give a criterion ensuring the inherent ordinary differential equations of the pair to be adjoint each to other. We describe the basis pairs in the invariant subspaces that yield adjoint pairs of essentially underlying ordinary differential equations. For a class of formally self-adjoint equations, we characterize the boundary conditions that lead to self-adjoint boundary value problems for the essentially underlying Hamiltonian systems.

Symmetry operators for Riemann’s method

Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in 2 variables. Chaundy’s equation, with 4 parameters, is the most general self-adjoint equation for which the Riemann function is known. Here we show that Chaundy’s equation possesses a two-dimensional vector space of second-order symmetry operators. Hence a new equivalence class of Riemann functions, admitting no first-order symmetries and obtainable only via a higher order symmetry, is found. A new 5 parameter Riemann function is then subsequently derived.

Complete solution of the Schr dinger equation of the complex manifoldCP2

Passing from the CP2 Lagrangian in Kahler form, to the Hamiltonian in terms of polar coordinates, this paper gives the complete set of solutions of the corresponding Schrodinger equation in a manner that makes fully explicit the SU(3) description of the energy eigenspaces. The solutions of the self-adjoint equation for the radial coordinates are derived most easily directly but also related to Jacobi polynomials.