We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials. To construct conservation laws, we develop and apply the most direct method which is effective to use in the case of two independent variables. Admitting possibility of dependence of conserved vectors on a number of potentials, we generalize the iteration procedure proposed by Bluman and Doran-Wu for finding nonlocal (potential) conservation laws. As an example, we completely classify potential conservation laws (including arbitrary order local ones) of diffusion--convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations.
We investigate conservation laws of diffusion-convection equations to construct first-order potential systems corresponding to these equations. We do two iterations of the construction procedure, looking, in the second step, for the first-order conservation laws of the potential systems obtained in the first step.
All possible linearly independent local conservation laws for $n$-dimensional diffusion--convection equations $u_t=(A(u))_{ii}+(B^i(u))_i$ were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O. Popovych, N.M. Ivanova, J. Math. Phys., 2005, V.46, 043502 (math-ph/0407008)] allows us to formulate the result in explicit closed form. Action of the symmetry groups on the conservation laws of diffusion equations is investigated and generating sets of con…
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding potential symmetries for the systems with multidimensional spaces of conservation laws. A class of variable coefficient (1+1)-dimensional nonlinear diffusion-convection equations of general form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$ is investigated. Using the most direct method, we carry out two classifications of local conservation laws up to equivalence rela…
We study local conservation laws of variable coefficient diffusion-convection equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. The main tool of our investigation is the notion of equivalence of conservation laws with respect to the equivalence groups. That is why, for the class under consideration we first construct the usual equivalence group $G^{\sim}$ and the extended one $\hat G^{\sim}$ including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group $\hat G^{\sim}$ has interesting structure since it contains a non-trivial subgroup…
We derive a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz-Ladik (AL) hierarchy. Our methods rely on a recursive approach to the AL hierarchy using Laurent polynomials and on asymptotic expansions of the Green's function of the AL Lax operator, a five-diagonal finite difference operator.
We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a byproduct, we obtain an infinite-dimensional differential covering over the Przanowski equation.
In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial co…
It is shown that equations of the Korteweg-de Vries hierarchy and their conservation laws can be expressed via the whole powers of an integro-differential operator and functions provided by them.
We propose a new truncation scheme for Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. We approximate the three particle distribution function $f_{3}(1,2,3,t)$ in terms of $f_{2}(1,2,t)$, $f_{1}(3,t)$ and two point correlation functions $\left\lbrace g_{2}(1,3,t), g_{2}(2,3,t)\right\rbrace $. Further $f_{2}$ is expressed in terms of $f_{1}(1,t)$ and $g_{2}(1,2,t)$ to close the hierarchy, resulting a set of coupled kinetic equations for $f_{1}$ and $g_{2}$. In this paper we show that, for velocity independent correlations, the kinetic equation for $f_{1}$ reduces to the model proposed by…
We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamil…
This paper is devoted to the negative flows of the AKNS hierarchy. The main result of this work is the functional representation of the extended AKNS hierarchy, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local systems of higher order, derive the generating function of the conservation laws and the N-dark-soliton solutions for the ex…
We construct an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation $A u_t\,u_{xy}+B u_x\,u_{ty}+C u_y\,u_{tx} = 0$, where $A,B,C$ are constants and $A+B+C\neq 0$, using a nonisospectral Lax pair. As a byproduct, we present new coverings for the ABC equation. The method of proof of nontriviality of the conservation laws under study is quite general and can be applied to many other integrable multidimensional systems.
This paper is devoted to the negative flows of the derivative nonlinear Schrödinger hierarchy (DNLSH). The main result of this work is the functional representation of the extended DNLSH, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local ones of higher order, derive the generating function of the conservation laws and the N-soliton s…
Two integrable $U(1)$-invariant peakon equations are derived from the NLS hierarchy through the tri-Hamiltonian splitting method. A Lax pair, a recursion operator, a bi-Hamiltonian formulation, and a hierarchy of symmetries and conservation laws are obtained for both peakon equations. These equations are also shown to arise as potential flows in the NLS hierarchy by applying the NLS recursion operator to flows generated by space translations and $U(1)$-phase rotations on a potential variable. Solutions for both equations are derived using a peakon ansatz combined with an oscillatory temporal p…
A computational classification of contact symmetries and higher-order local symmetries that do not commute with $t,x$, as well as local conserved densities that are not invariant under $t,x$ is carried out for a generalized version of the Krichever-Novikov equation. Several new results are obtained. First, the Krichever-Novikov equation is explicitly shown to have a local conserved density that contains $t,x$. Second, apart from the dilational point symmetries known for special cases of the Krichever-Novikov equation and its generalized version, no other local symmetries with low differential …
The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Fractional kinetic equation for the system of charged particles are considered.
