Nonlinear self-adjointness through differential substitutions

It is known (Ibragimov, 2011; Galiakberova and Ibragimov, 2013) [14,18] that the property of nonlinear self-adjointness allows to associate conservation laws of the equations under study, with their symmetries. In this paper we show that, even when the equation is nonlinearly self-adjoint with a non differential substitution, finding the explicit form of the differential substitution can provide new conservation laws associated to its symmetries. By using the general theorem on conservation laws (Ibragimov, 2007) [11] and the property of nonlinear self-adjointness we find some new conservation laws for the modified Harry-Dym equation. By using a differential substitution we construct a conservation law for the Harry-Dym equation, which has not been derived before using Ibragimov method.     

Conservation laws for two-phase filtration models

The paper is devoted to investigation of group properties of a one-dimensional model of two-phase filtration in porous medium. Along with the general model, some of its particular cases widely used in oil-field development are discussed. The Buckley–Leverett model is considered in detail as a particular case of the one-dimensional filtration model. This model is constructed under the assumption that filtration is one-dimensional and horizontally directed, the porous medium is homogeneous and incompressible, the filtering fluids are also incompressible. The model of “chromatic fluid” filtration is also investigated. New conservation laws and particular solutions are constructed using symmetries and nonlinear self-adjointness of the system of 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.     

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.     

Self-adjointness of a generalized Camassa-Holm equation

It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, it is shown in [1] and [2] that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization [3] of the Camassa-Holm equation [4], we prove that the Camassa-Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa-Holm equation using its symmetries.     

On the conservation laws and invariant solutions of the mKdV equation

In this paper, we consider modified Korteweg-de Vries (mKdV) equation. By using the nonlocal conservation theorem method and the partial Lagrangian approach, conservation laws for the mKdV equation are presented. It is observed that only nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws. In addition, invariant solution is obtained by utilizing the relationship between conservation laws and Lie-point symmetries of the equation.     

Conservation laws for a modified lubrication equation

In this work we study the conservation laws of a modified lubrication equation, which describes the dynamics of the interfacial motion in phase transition. We show that the equation is nonlinear self-adjoint and has an exact Lagrangian with an auxiliary function. As a result, by a general theorem on conservation laws proved by Nail Ibragimov recently and Noether’s theorem, some new conservation laws for the equation are obtained. Our results show that the non-locally defined conservation laws generated by Noether’s theorem are equivalent to the local ones given by Ibragimov’s theorem.     

Conservation laws for the Black–Scholes equation

In the search for solutions to the important partial differential equation due to Black, Scholes and Merton potential symmetries are very useful as new solutions of the equation can be obtained as a result. These potential symmetries require that the equation be written in conserved form, ie. we need to determine conservation laws for the equation. We calculate the conservation laws utilizing the point symmetries of the equation following the method of Kara and Mahomed [A.H. Kara, F.M. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40].     

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 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.     

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).     

Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation

In this paper, we consider a class of generalized diffusion equations which are of great interest in mathematical physics. For some of these equations that model fast diffusion, nonclassical and nonclassical potential symmetries are derived. These symmetries allow us to increase the number of solutions. These solutions are unobtainable neither from classical nor from classical potential symmetries. Copyright © 2007 John Wiley & Sons, Ltd.     

On the exact solutions and conservation laws to the Benjamin-Ono equation

In the present work, we dealt with exact solutions and conservation laws of the Benjamin-Ono equation. We obtained exact solutions of given equation via the exp$(-\Phi (\xi ))$ method. The obtained solutions are included the hyperbolic functions, trigonometric functions and rational functions. By using the multiplier approach, the conservation laws of the mentioned equation was founded.     

New conservation laws obtained directly from symmetry action on a known conservation law

Two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries. The first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation. When the contact transformation is a symmetry of the given system, then the corresponding conservation law is a conservation law of the given system. The second formula checks a priori whether or not the action of a symmetry (continuous or discrete) on a conservation law can yield one or more new conservation laws of the given system. Several examples are considered, including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws.     

Type II hidden symmetries through weak symmetries for some wave equations

The Type II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie-point symmetry. In [Gandarias RML. Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations. J Math Anal Appl 2008;348:752–9] it was shown that the provenance of the Type II Lie point hidden symmetries found for differential equations can be explained by considering weak symmetries or conditional symmetries of the original PDE.In this paper we analyze the connection between one of the methods analyzed in [Abraham-Shrauner B, Govinder KS. Provenance of Type II hidden symmetries from nonlinear partial differential equations. J Nonlin Math Phys 2006;13:612–22] and the weak symmetries of some partial differential equations in order to determine the source of these hidden symmetries. We have considered some of the models presented in [Abraham-Shrauner B, Govinder KS. Provenance of Type II hidden symmetries from nonlinear partial differential equations. J Nonlin Math Phys 2006;13:612–22], as well as the linear two-dimensional and three-dimensional wave equations [Abraham-Shrauner B, Govinder KS, Arrigo JA. Type II hidden symmetries of the linear 2D and 3D wave equations. J h Phys A Math Theor 2006;39:5739–47].     

Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations

The Type-II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. In [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622] Abraham-Shrauner and Govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries. The Lie point symmetries of a model equation and the two-dimensional Burgers' equation and their descendants were used to identify the hidden symmetries. In this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries. We have considered the same models presented in [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622], as well as the WDVV equations of associativity in two-dimensional topological field theory which reduces, in the case of three fields, to a single third order equation of Monge-Ampère type. We have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using Lie symmetries, however when is reduced by using nonclassical symmetries the reduced partial differential equation gains Lie symmetries.     

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.     

1-Soliton solution and conservation laws for nonlinear wave equation in semiconductors

This paper obtains the 1-soliton solution of a nonlinear wave equation that arises in the study of semiconductors. The conserved quantities are also calculated from this equation. Furthermore, additional non-trivial conserved quantities are computed using the invariance and multiplier approach based on the well known result that the Euler-Lagrange operator annihilates the total divergence.