Quasi Self-adjoint Research Articles

Lie analysis, conserved quantities and solitonic structures of Calogero-Degasperis-Fokas equation

The paper investigates Calogero-Degasperis-Fokas (CDF) equation, an exactly solvable third order nonlinear evolution equation (Fokas, 1980). All possible functions for the unknown function F(ν) in the considered equation are listed that contains the nontrivial Lie point symmetries. Furthermore, nonlinear self-adjointness is considered and for the physical parameter A≠0 the equation is proved not strictly self-adjoint equation but it is quasi self-adjoint or more generally nonlinear self-adjoint equation. In addition, it is remarked that CDF equation admits a minimal set of Lie algebra under invariance test of Lie groups. Subsequently, Lie symmetry reductions of CDF equation are described with the assistance of an optimal system, which reduces the CDF equation into different ordinary differential equations. Besides, Lie symmetries are used to indicate the associated conservation laws. Also, the well-known (G′/G)-expansion approach is applied to obtain the exact solutions. These new periodic and solitary wave solutions are feasible to analyse many compound physical phenomena in the field of sciences.

Self-Adjointness and Conservation Laws of Frobenius Type Equations

The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra Z2ε in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws.

Comment on the paper “On conservation laws by Lie symmetry analysis for [formula omitted]-dimensional Bogoyavlensky–Konopelchenko equation in wave propagation” by S. Saha Ray

In a recent paper referred to in the title, the author used the concept of quasi self-adjointness to obtain conservation laws for a system of the (2+1)-dimensional Bogoyavlensky–Konopelchenko equation. Apart from the adjoint system of equations, all the results on the quasi self-adjointness and the subsequent conservation laws obtained are inaccurate. In this comment, we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through Noether theorem.

On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation in wave propagation

In this paper, using the Lie group analysis method, the infinitesimal generators for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained. The new concept of nonlinear self-adjointness of differential equations is used for construction of nonlocal conservation laws. The conservation laws for the (2+1)-dimensional Bogoyavlensky–Konopelchenko equation are obtained by using the new conservation theorem method and the formal Lagrangian approach. Transforming this equation into a system of equations involving with two dependent variables, it has been shown that the resultant system of equations is quasi self-adjoint and finally the new nonlocal conservation laws are constructed by using the Lie symmetry operators.

Dynamics of solitons to the ill-posed Boussinesq equation

In this paper, we analyze the dynamic behavior of the ill-posed Boussinesq equation (IPBE) that arises in nonlinear lattices and also in shallow water waves. Some solitary wave solutions are obtained by using the solitary wave ansatz method and the Bernoulli sub-Ode. By applying the technique of nonlinear self-adjoint, a quasi self-adjoint substitution for the IPBE is constructed. The classical symmetries of the equation are constructed. Then, we used along with the obtained nonlinear self-adjoint substitution to construct a set of new conservation laws (Cls).

On Symmetry Analysis and Conservation Laws of the AKNS System

Abstract The Lie symmetry analysis is applied to study the Ablowitz–Kaup–Newell–Segur (AKNS) system of water wave model. The AKNS system can be obtained from a dispersive-wave system via a variable transformation. Lie point symmetries and corresponding point transformations are determined. The optimal system of one-dimensional subalgebras is presented. On the basis of the optimal system, the similarity reductions and the invariant solutions are obtained. Some conservation laws are derived using the multipliers. In addition, the AKNS system is quasi self-adjoint. The conservation laws associated with the symmetries are also constructed.

Nonlinear Self-Adjointness and Generalized Conserved Quantities of the Inviscid Burgers’ Equation with Nonlinear Source

The conservation laws of the inviscid Burgers’ equation under consideration have been studied recently using the concept of quasi self-adjointness and self-adjointness. These two concepts have been extended to the notion of nonlinear self-adjointness to enable more conservation laws of differential equations, that are not achievable through them, be constructed. We explore this avenue in the present study and establish the generalized nonlinearly self-adjoint condition for the inviscid Burgers’ equation. This condition not only gives rise to new nontrivial independent conserved vectors, but also includes results of previous work as a particular case.

Self-adjointness, Group Classification and Conservation Laws of an Extended Camassa-Holm Equation

In this paper, we prove that equation 2 ( ) 2 3 = 0 t x t x x x x E ≡ u −u + u f u − au u − buu is self-adjoint and quasi self-adjoint, then we construct conservation laws for this equation using its symmetries. We investigate a symmetry classification of this nonlinear third order partial differential equation, where f is smooth function on u and a, b are arbitrary constans. We find Three special cases of this equation, using the Lie group method.

Nonlinear self-adjointness and conservation laws of Klein–Gordon–Fock equation with central symmetry

The concept of nonlinear self-adjointness, introduced by Ibragimov, has significantly extends approaches to constructing conservation laws associated with symmetries since it incorporates the strict self-adjointness, the quasi self-adjointness as well as the usual linear self-adjointness. Using this concept, the nonlinear self-adjointness condition for the Klein–Gordon–Fock equation was established and subsequently used to construct simplified but infinitely many nontrivial and independent conserved vectors. The Noether’s theorem was further applied to the Klein–Gordon–Fock equation to explore more distinct first integrals, result shows that conservation laws constructed through this approach are exactly the same as those obtained under strict self-adjointness of Ibragimov’s method.

Conservation laws of inviscid Burgers equation with nonlinear damping

In this paper, the new conservation theorem presented in Ibragimov (2007) [14] is used to find conservation laws of the inviscid Burgers equation with nonlinear damping ut+g(u)ux+λh(u)=0. We show that this equation is both quasi self-adjoint and self-adjoint, and use these concepts to simplify conserved quantities for various choices of g(u) and h(u).

On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models

In this paper we consider a class of evolution equations up to fifth-order containing many arbitrary smooth functions from the point of view of nonlinear self-adjointness. The studied class includes many important equations modeling different phenomena. In particular, some of the considered equations were studied previously by other researchers from the point of view of quasi self-adjointness or strictly self-adjointness. Therefore we find new local conservation laws for these equations invoking the obtained results on nonlinearly self-adjointness and the conservation theorem proposed by Nail Ibragimov.

Quasi self-adjointness of a class of third order nonlinear dispersive equations

In this paper we find the conditions of self-adjointness and quasi self-adjointness for a class of third order PDEs. As an application, in some cases, we get conservation laws.

Conservation laws for a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model

We find the Lie point symmetries of a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model. Then we establish its quasi self-adjointness and corresponding conservation laws.

Self-adjointness and conservation laws of two variable coefficient nonlinear equations of Schrödinger type

In this paper, some recent concepts and results on self-adjointness and conservation laws are applied to two variable coefficient nonlinear equations of Schrödinger type: the generalized variable coefficient nonlinear Schrödinger (GVCNLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients. The two equations are changed to two real systems by a proper transformation. To obtain the formal Lagrangians of the two systems, we discuss their self-adjointness and find that the GVCNLS system is weak self-adjoint and the CQNLS system is quasi self-adjoint. Having performed Lie symmetry analysis for the two systems, we find five nontrivial conservation laws for the GVCNLS system and four nontrivial conservation laws for the CQNLS system by using a general theorem on conservation laws given by Ibragimov.

Conservation laws for a class of quasi self-adjoint third order equations

In this work we consider a class of third-order nonlinear partial differential equation containing two un-specified coefficient functions of the dependent variable which include various integrable and nonintegrable equations. We determine the subclasses of these equations which are self-adjoint and quasi self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov we find conservation laws for some of these partial differential equations without classical Lagrangians.

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.

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-adjointness and conservation laws of a generalized Burgers equation

A (2 + 1)-dimensional generalized Burgers equation is considered. Having written this equation as a system of two dependent variables, we show that it is quasi self-adjoint and find a nontrivial additional conservation law.