Conservation laws for nonlinear telegraph equations

A complete conservation law classification is given for nonlinear telegraph (NLT) systems with respect to multipliers that are functions of independent and dependent variables. It turns out that a very large class of NLT systems admits four nontrivial local conservation laws. The results of this work are summarized in tables which display all multipliers, fluxes and densities for the corresponding conservation laws. A physical example is considered for possible applications.     

On Lie symmetry analysis,conservation laws and solitary waves to a longitudinal wave motion equation

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.     

Group classification and conservation laws of nonlinear filtration equation with a small parameter

Group classification of the perturbed nonlinear filtration equation is performed assuming that the perturbation is an arbitrary function of the dependent variable. The nonlinear self-adjointness of the equation under consideration is investigated. Using these results, the approximate conservation laws are constructed.     

Symmetry reduction, exact solutions and conservation laws of the Sawada-Kotera-Kadomtsev-Petviashvili equation

In this paper, by applying a direct symmetry method, we obtain the symmetry reduction, group invariant solution and many new exact solutions of SK-KP equation, which include Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and so on. At last, we also give the conservation laws of SK-KP equation.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

Appell’s lemma and conservation laws of KdV equation

An alternative method to construct a class of conservation laws of the KdV equation based on the classical Appell’s lemma and the trace formulas of Deift-Trubowitz type is studied. A new type of infinite sequence of conservation laws whose local densities cannot be expressed in terms of differential polynomials is constructed.     

Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities

A class of variable coefficient (1+1)-dimensional nonlinear reaction–diffusion equations of the general form f(x)u_t=(g(x)uⁿu_x)_x+h(x)u^m is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.     

Scalar conservation laws with stochastic forcing

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.     

Group properties and conservation laws for nonlocal shallow water wave equation

Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.     

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.     

Exact solutions and conservation laws of Zakharov-Kuznetsov modified equal width equation with power law nonlinearity

This paper obtains solutions to the Zakharov-Kuznetsov modified equal width equation with power law nonlinearity. The Lie symmetry approach and the simplest equation method are used to obtain these solutions. Moreover, conservation laws are derived for the underlying equation by employing two approaches: the new conservation theorem and the multiplier method.     

On the optimization of the initial boundary value problem for a conservation law

In the present note, the theory of shift differentiability for the Cauchy problem is extended to the case of an initial boundary value problem for a conservation law. This result allows to exhibit an Euler-Lagrange equation to be satisfied by the extrema of integral functionals defined on the solutions of initial boundary value problems of this kind.     

Solution of convex conservation laws in a strip

In this paper we consider scalar convex conservation laws in one space variable in a stripD =(x, t): 0 ≤x ≤1,t > 0 and obtain an explicit formula for the solution of the mixed initial boundary value problem, the boundary data being prescribed in the sense of Bardos-Leroux and Nedelec. We also get an explicit formula for the solution of weighted Burgers equation in a strip.     

Explicit generalized solutions to a system of conservation laws

This paper studies a special 3 by 3 system of conservation laws which cannot be solved in the classical distributional sense. By adding a viscosity term and writing the system in the form of a matrix Burgers equation an explicit formula is found for the solution of the pure initial value problem. These regularized solutions are used to construct solutions for the conservation laws with initial conditions, in the algebra of generalized functions of Colombeau. Special cases of this system were studied previously by many authors.     

Exact solutions and conservation laws of (2 + 1)-dimensional Boiti-Leon-Pempinelli equation

In this paper, we obtain the symmetry group theorem by using the modified CK’s direct method, and some new exact solutions of (2 + 1)-dimensional BLP equation. Also we derive the corresponding Lie algebra and the conservation laws of BLP equation.     

Convergence of a continuous BGK model for conservation laws on the quarter plane

We consider a scalar conservation law in the quarter plan. This equation is approximated in a kinetic BGK model with infinite set of velocities. The convergence is established in the general BV framework, without special restrictions on the flux nor on the equilibrium problem's data.     

Conservation laws with a random source

We study the scalar conservation law with a noisy nonlinear source, namely,u _l + f(u)_x = h(u, x, t) + g(u)W(t), whereW(t) is the white noise in the time variable, and we analyse the Cauchy problem for this equation where the initial data are assumed to be deterministic. A method is proposed to construct approximate weak solutions, and we then show that this yields a convergent sequence. This sequence converges to a (pathwise) solution of the Cauchy problem. The equation can be considered as a model of deterministic driven phase transitions with a random perturbation in a system of two constituents. Finally we show some numerical results motivated by two-phase flow in porous media. This research has been supported by VISTA (a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap, Statoil) and NAVF (the Norwegian Research Council for Science and the Humanities).