In this work we consider the problem on group classification and conservation laws of the general first order evolution equations. We obtain the subclasses of these general equations which are quasi-self-adjoint and self-adjoint. By using the recent Ibragimov's Theorem on conservation laws, we establish the conservation laws of the equations admiting self-adjoint equations. We illustrate our results applying them to the inviscid Burgers' equation. In particular an infinite number of new symmetries of these equations are found and their corresponding conservation laws are established.
Nonlinear self-adjointness method for constructing conservation laws of partial differential equations (PDEs) is further studied. We show that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and vice versa. Consequently, each symmetry of PDEs corresponds to a conservation law via a formula if the system of PDEs is nonlinearly self-adjoint with differential substitution. As a byproduct, we find that the set of differential substitutions includes the set of conservation law multipliers as a subset. The results are illustrated by three typical examples.
Conservation Laws of Multidimensional Diffusion-Convection Equations
利用直接方法和复合变分原理构造了$n$维扩散-对流方程$u_t=(A(u))_{ii}+(B^i(u))_i$的所有可能的线性无关局部守恒律.守恒律分类法在点变换群上的应用[R. O. Popovych，新墨西哥州Ivanova，J. Math. Phys.，2005，V.46，043502（math-ph/0407008）]允许我们以显式封闭形式来公式化结果。研究了对称群对扩散方程守恒律的作用，构造了守恒律的生成元集。
A complete classification of all low-order conservation laws is carried out for a system of coupled semilinear wave equations which is a natural two-component generalization of the nonlinear Klein-Gordon equation. The conserved quantities defined by these conservation laws are derived and their physical meaning is discussed.
Conservation laws are formulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not require the existence of a Lagrangian for a given system, and the presented examples include computations of conserved densities for the heat equation, Burgers' equation and the Korteweg-de Vries equation.
It is shown that expected values of free scalar and vector quantum fields on graphs satisfy the same conservation laws as the classic fields. It is demonstrated that a modified version of conservation laws is satisfied for Villain action. The proofs are based on the consideration of multidimensional normal distribution.
A new approach, combining the Ibragimov method and the one by Anco and Bluman, with the aim of algorithmically computing local conservation laws of partial differential equations, is discussed. Some examples of the application of the procedure are given. The method, of course, is able to recover all the conservation laws found by using the direct method; at the same time we can characterize which symmetry, if any, is responsible for the existence of a given conservation law. Some new local conservation laws for the Short Pulse equation and for the Fornberg Whitham equation are also determined.
In the present paper we consider nonlinear multidimensional Cahn-Hilliard and Kuramoto-Sivashinsky equations that have many important applications in physics and chemistry, and a certain natural generalization of these equations. For an arbitrary number of spatial independent variables we present a complete list of cases when the generalized Cahn-Hilliard-Kuramoto-Sivashinsky equation admits nontrivial local conservation laws of any order, and for each of those cases we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under stu…
The methods of abstract simplicial homology and cohomology are reviewed and applied to the topology of electrical networks. Kirchhoffs laws of electrical circuits are shown to be manifestly homological in their origins. Since they are based in conservation laws, the geometric realization of abstract simplicial complexes is then reviewed and applied to the case of mechanical networks. The equilibrium condition for statics, the conservation laws for closed systems, and the balance principles for open systems are then shown to admit homological formulations.
A complete classification of low-order conservation laws is obtained for time-dependent generalized Korteweg-de Vries equations. Through the Hamiltonian structure of these equations, a corresponding classification of Hamiltonian symmetries is derived. The physical meaning of the conservation laws and the symmetries is discussed.
A conservation law theorem stated by N. Ibragimov along with its subsequent extensions are shown to be a special case of a standard formula that uses a pair consisting of a symmetry and an adjoint-symmetry to produce a conservation law through a well-known Fréchet derivative identity. Also, the connection of this formula (and of Ibragimov's theorem) to the standard action of symmetries on conservation laws is explained, which accounts for a number of major drawbacks that have appeared in recent work using the formula to generate conservation laws. In particular, the formula can generate trivia…
We characterize all cases when a certain natural generalization of the Infeld--Rowlands equation admits nontrivial local conservation laws of any order, and give explicit form of these conservation laws modulo trivial ones. Furthermore, we prove that the equation under study admits no nontrivial local Hamiltonian and symplectic structures and no nontrivial local Noether and inverse Noether operators; the method of establishing the said nonexistence results can be readily applied to many other PDEs.
Lie point symmetries of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates are considered. Complete Lie group classification of these equations reduced to a scalar second-order PDE is performed. The classification parameter is the entropy. Noether theorem is applied for constructing conservation laws. The conservation laws can be represented in the gas dynamics variables. For the basic adiabatic case invariant and conservative difference schemes are discussed.
A recent paper considered symmetries and conservation laws of the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. This paper analyses the one-dimensional magnetohydrodynamics flows with cylindrical symmetry in the mass Lagrangian coordinates. The medium is assumed inviscid and thermally non-conducting. It is modeled by a polytropic gas. Symmetries and conservation laws are found. The cases of finite and infinite electric conductivity need to be analyzed separately. For finite electric conductivity $σ(ρ,p)$ we perform Lie group classification, which iden…
Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Using group classification and Noether's theorem, conservation laws are obtained. Their counterparts in Eulerian coordinates are given. Among these counterparts there are new conservation laws.
In this paper, we construct and analyse the symmetries and conservation laws (conserved densities) of a model of a nonlinear Scrodinger equation with PT-symmetric potentials and inhomogeneity.
We prove that the generalized Swift--Hohenberg equation with nonlinear right-hand side, a natural generalization of the Swift--Hohenberg equation arising in physics and describing inter alia pattern formation, has no nontrivial local conservation laws.
We provide a complete classification of generalized and formal symmetries and local conservation laws for an evolution equation which generalizes the Kawahara equation having important applications in the study of plasma waves and capillary-gravity water waves. In particular, we show that the equation under study admits no genuinely generalized symmetries and has only finitely many local conservation laws, and thus this equation is not symmetry integrable.
Extensivity of conservation laws of the quantum Hirota model on a $1+1$ dimensional lattice is considered. This model can be interpreted in terms of an integrable many-body quantum Floquet dynamics. We establish the procedure to generate a continuous family of quasilocal conservation laws from the conserved operators proposed by Faddeev and Volkov. The Hilbert-Schmidt kernel which allows the calculation of inner products of these new conservation laws is explicitly computed. This result has potential applications in quantum quench and transport problems in integrable quantum field theories.
We find noncommutative analogs for well-known polynomial evolution systems with higher conservation laws and symmetries. The integrability of obtained non-Abelian systems is justified by explicit zero curvature representations with spectral parameter.
Using the lax pair, nonlocal symmetry of the coupled Hirota equation is obtained. By introducing an appropriate auxiliary dependent variable the nonlocal symmetry is successfully localized to a Lie point symmetry. For the closed prolongation, one-dimensional optimal systems and nonlocal conservation laws of the coupled Hirota equation are studied.
We consider a new nonlocal formulation of the water-wave problem for a free surface with an irrotational flow based on the work of Ablowitz, Fokas, and Musslimani and presented in the recent work of Oliveras. The main focus of the short paper is to show how one can systematically derive Olver's eight conservation laws not only for an irrotational fluid, but also for constant vorticity (linear shear flow) without explicitly relying on the underlying Lie symmetries. This allows us to make draw new conclusions about conservation laws and posit the existence of additional, nonlocal, conservation l…
Using the concept of variational tricomplex endowed with a presymplectic structure, we formulate the general notion of symmetry. We show that each generalized symmetry of a gauge system gives rise to a sequence of conservation laws that are represented by on-shell closed forms of various degrees. This extends the usual Noether's correspondence between global symmetries and conservation laws to the case of lower-degree conservation laws and not necessarily variational equations of motion. Finally, we equip the space of conservation laws of a given degree with a Lie bracket and establish a homom…
For systems of partial differential equations in three spatial dimensions, dynamical conservation laws holding on volumes, surfaces, and curves, as well as topological conservation laws holding on surfaces and curves, are studied in a unified framework. Both global and local formulations of these different conservation laws are discussed, including the forms of global constants of motion. The main results consist of providing an explicit characterization for when two conservation laws are locally or globally equivalent, and for when a conservation law is locally or globally trivial, as well as…
