Systems Of Mathematical Physics Research Articles

Symmetric form of the magnetohydrodynamics equations

It is well known that many linear hyperbolic systems of mathematical physics are symmetric hyperbolic systems. In the case of nonlinear equations, the question of whether they can be written in symmetric form is far from simple.

Canonical Equations of Hamilton with Symmetry and Their Applications

Two systems of mathematical physics are defined by us, which are the first-order differential system (FODS) and the second-order differential system (SODS). Basing on the conventional Legendre transformation, we obtain a new kind of canonical equations of Hamilton (CEH) with some kind of symmetry. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs, on basis of the same conventional Legendre transformation. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way. Based on the new CEH, the approximate soliton solution of the nonlocal nonlinear Schrödinger equation is obtained, and the soliton stability is analysed analytically as well. Furthermore, because the symmetry of a system is closely connected with certain conservation theorem of the system, the new CEH may be useful in some complicated systems when the symmetry considerations are used.

Application of Jordan canonical form and symplectic matrix in fractional differential models

Under consideration of this paper is the application of Jordan canonical form and symplectic matrix to two conformable fractional differential models. One is the new conformable fractional vector conduction equation which is reduced by using the Jordan canonical form of coefficient matrix and solved exactly, and the other is the new conformable fractional vector dynamical system with Hamilton matrix and symplectic matrix, which is derived by constructing the conformable fractional Euler-Lagrange equation and using fractional variational principle. It is shown that Jordan canonical form and symplectic matrix can be used to deal with some other conformable fractional differential systems in mathematical physics.

Explicit solutions of some equations and systems of mathematical physics

This paper deals at first with a fully integrable evolution system of nonlinear partial differential equations (PDEs) which is a generalization of the classical Heisenberg ferromagnet equation. Then the scalar variant of this system is considered. Looking for solutions of special form, the problem of finding explicit solutions of the above-mentioned equations is reduced to the global solvability of overdetermined real-valued systems of nonlinear PDEs. In many cases particular solutions which are not solitons are expressed by classical functions including some special ones as Jacobi elliptic functions, Legendre elliptic functions, and Weierstrass normal elliptic integrals. A geometrical visualization of several solutions is also proposed.

Dissipative measure-valued solutions for general conservation laws

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued – strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system et al. and there are also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems, and most importantly, we give examples of equations, for which the aspect of measure-valued – strong uniqueness has not been considered before, like incompressible magnetohydrodynamics and shallow water magnetohydrodynamics.

Symmetry Operators and Solutions to Differential Equations in Algebra

The symmetry operator method for finding an analytical form for the solutions to a given PDE system is presented. The main tools for accomplishing this aim include computation and grading of the Lie algebra infinitesimals of symmetry groups admitted by a given PDE system, establishing fusion rules, and then checking the commutator relations after performing the grading. We discuss several examples of the differential systems of mathematical physics based on the splitting of heat and the Euler–Tricomi, generalized Cauchy–Riemann and Dirac equations in non-associative algebras.

A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation

A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such cases most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws; they are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake the role of physical admissibility conditions for weak solutions. We want to answer the question: what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality? An archetypal example of such a result was derived for the incompressible Euler system in the context of Onsager’s conjecture in the early nineties. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.

Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics

In this study, a new method called improved Bernoulli sub-equation function method has been proposed. This method is based on the Bernoulli sub-ODE method. After we mention the general properties of proposed method, we apply this algorithm to the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation system. This gives us some new prototype solutions such as exponential and rational function solutions. Then, we have plotted two- and three-dimensional surfaces of analytical solutions. Finally, we have submitted a comprehensive conclusion.

Canonical equations of Hamilton for the nonlinear Schrödinger equation

We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.

The new approach of the generalized (G′/G)-expansion method for nonlinear evolution equations

In this article, we apply the new generalized (G′/G)-expansion method to solve two nonlinear partial differential equations and construct traveling wave solutions in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. We underline the power of the new generalized (G′/G)-expansion method in providing generalized solitary wave solutions of different physical structures. It is shown that the new generalized (G′/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear partial differential equation systems in mathematical physics and engineering.

FINITE SYMMETRY GROUP AND COHERENT SOLITON SOLUTIONS FOR THE BROER–KAUP–KUPERSHMIDT SYSTEM

A modified CK direct method is generalized to find finite symmetry groups of nonlinear mathematical physics systems. For the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, both the Lie point symmetry and the non-Lie symmetry groups are obtained by this method. While using the traditional Lie approach, one can only find the Lie symmetry groups. Furthermore, abundant localized structures of the BKK equation are also obtained from the non-Lie symmetry group.

Iterative analytic approximation to nonlinear convection dominated systems

In this paper we are interested in the analytic approximation of nonlinear convection dominated systems of mathematical physics. Solutions of such systems, contaminated by a small parameter, often show sharp boundary and interior layers. The problem becomes still more complex as the equation under consideration has nonlinearity together with small dissipation. Then, shock waves appear alongside boundary layers. To approximate the multi-scale solution of convection dominated problems, we present and analyze an iterative analytic method based on a Lagrange multiplier technique. The Lagrange multiplier is obtained optimally, in a general setting, using variational theory and Liouville–Green transforms. The idea of the paper is to overcome the well known difficulties associated with the numerical methods. Examples, with quadratic nonlinear convection terms and quasi-linear terms, are taken into account to show the effectiveness and accuracy of the present approach. It is observed that the method is straightforward, highly accurate, brief and can also be functional to other nonlinear evolution equations of mathematical physics.

Explicit solutions of nonlinear wave equation systems

We apply the (G′/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. We highlight the power of the (G′/G)-expansion method in providing generalized solitary wave solutions of different physical structures. It is shown that the (G′/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.

On Bers Generating Functions for First Order Systems of Mathematical Physics

Considering one of the fundamental notions of Bers’ theory of pseudoanalytic functions the generating pair via an intertwining relation we introduce its generalization for biquaternionic equations corresponding to different first-order systems of mathematical physics with variable coefficients. We show that the knowledge of a generating set of solutions of a system allows one to obtain its different form analogous to the complex equation describing pseudoanalytic functions of the second kind and opens the way for new results and applications of pseudoanalytic function theory. As one of the examples the Maxwell system for an inhomogeneous medium is considered, and as one of the consequences of the introduced approach we find a relation between the time-dependent one-dimensional Maxwell system and hyperbolic pseudoanalytic functions and obtain an infinite system of solutions of the Maxwell system. Other considered examples are the system describing force-free magnetic fields and the Dirac system from relativistic quantum mechanics.

Group foliation of the Lamé equations of the classical dynamical theory of elasticity

We perform the group foliation of the system of Lame equations of the classical dynamical theory of elasticity for an infinite subgroup contained in a normal divisor of the main group. The resolving system of this foliation includes the following two classical systems of mathematical physics: the system of equations of vortex-free acoustics and the system of Maxwell equations, which allows one to use wider groups to obtain exact solutions of the Lame equations. We obtain a first-order conformal-invariant system, which describes shear waves in a three-dimensional elastic medium. We also give examples of partially invariant solutions.

Travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov ([formula omitted]) system

In this paper, travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov system (called D( m,n) system) are studied by using the Weierstrass elliptic function method. As a result, more new exact travelling wave solutions to the D( m,n) system are obtained including not only all the known solutions found by Xie and Yan but also other more general solutions for different parameters m,n. Moreover, it is also shown that the D( m,1) system with linear dispersion possess compacton and solitary pattern solutions. Besides that, it should be pointed out that the approach is direct and easily carried out without the aid of mathematical software if compared with other traditional methods. We believe that the method can be widely applied to other similar types of nonlinear partial differential equations (PDEs) or systems in mathematical physics.

Lie Symmetry Groups of (2+1)-Dimensional BKP Equation and Its New Solutions

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry. Furthermore, the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.

A Maple Package to Compute Lie Symmetry Groups and Symmetry Reductions of (1+1)-Dimensional Nonlinear Systems

Armed with the computer algebra system Maple, using a direct algebraic substitution method, we obtain Lie point symmetries, Lie symmetry groups and the corresponding symmetry reductions of one component nonlinear integrable and nonintegrable equations only by clicking the ‘Enter’ key. Abundant (1+1)-dimensional nonlinear mathematical physical systems are analysed effectively by using a Maple package LieSYMGRP proposed by us.

Lie symmetry groups of high dimensional non-integral nonlinear systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well known (3+1)-dimensional KP equation and the (2+1)-dimensional KdV equation. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.

Non-Lie Symmetry Groups of (2+1)-Dimensional NonlinearSystems

A modified direct method is developed to find finite symmetrygroups of nonlinear mathematical physics systems. Applying themodified direct method to the well-known (2+1)-dimensionalasymmetric Nizhnik–Novikov–Vesselovequation and Nizhnik–Novikov–Vesselov equation,both the Lie point symmetry groupsand the non-Lie symmetry groups are obtained. The Lie symmetrygroups obtained via traditional Lie approaches are only specialcases. Furthermore, the expressions of the exact finitetransformations of the Lie groups are much simpler than thoseobtained via the standard approaches.