General Families Of Solutions Research Articles

The Cahn-Hilliard Equation – the application of the exponential transform method leading to a new insight for interfaces

The Cahn-Hilliard Equation was introduced to study phase separation in binary alloy glasses and polymers and is a good approach to spinodal decomposition. General families of solutions are of basic interest. Therefore a complete characterization of the group properties is given. We determine the Lie point symmetry vector fields and calculate similarity solutions. Depending upon the approach, here, in this case, the traveling-wave reduction leads to new nonlinear ordinary differential equations which only can be solved by numerical methods. We show that these ordinary differential equations are not of Painleve type and can therefore not fully be integrated. The crucial point of the paper is the fact that we are able to calculate new classes of solutions by a new algebraic method derived recently by the author. These new solutions, important in relevant applications, are discussed whereby we support the theoretical results by illustrative figures.

A note on new solitary and similarity class of solutions of a fourth-order non-linear evolution equation

In this paper, Lie’s method is used to calculate new class of solutions of a less studied fourth-order non-linear partial differential equation. In our previous paper [A. Huber, Appl. Math. Comp. 166/2 (2005) 464], we have applied the tanh-method to generate solutions. In this case, special class of solutions in form of traveling waves result (single soliton solutions as well as class of irregular solutions). Therefore, general families of solutions are of basic interest. Doing so, new results can be found in [A. Huber, Appl. Math. Comp., in press] using the Weierstraß transformation derived first by the author. Moreover, a complete characterization of the group properties is given because group properties are unknown up to now. We determine the Lie point symmetry vector fields and calculate similarity “ansätze”. Further, we also derive a few non-linear transformations and some similarity solutions are obtained explicitly. The main purpose for the application of Lie’s method is of course the fact that we are able to calculate class of general solutions which do not underlie such strong restrictions as in the case of traveling wave “ansätze”. Otherwise, it is necessary to perform a group analysis in order to improve the solution manifold by an alternative way. In addition, the studied equation passes the Painlevé-test and seems to be integrable. General class of solutions in terms of elliptic functions are derived via Lie’s approach for the first time representing exact solitary wave propagation and surprisingly, cusp–solitary class of solutions are obtained.

A note on class of traveling wave solutions of a non-linear third order system generated by Lie’s approach

Abstract In this paper, Lie’s method is used to calculate solutions of a third order non-linear system of partial differential equations (nPDE). In our previous paper [Huber A. Appl Math Comput 2005;166/2:464], we have applied the tanh-method to generate solutions, in this case special class of solutions in form of traveling wave results (single soliton solutions as well as class of irregular solutions). Therefore, general families of solutions are of basic interest. Moreover, a complete characterization of the group properties is given. We determine the Lie point symmetry vector fields and calculate similarity “ansatze” for the first time. Further, we also derive a few non-linear transformations and some similarity solutions are obtained. The main purpose for the application of Lie’s method is of course the fact that we are able to calculate class of general solutions which do not underlie such strong restrictions as in the case of traveling wave “ansatze”. Otherwise, it is necessary to perform a group analysis in order to improve the solution manifold by an alternative way. Moreover, a criterion for the integrability via the Painleve-conjecture is given and further, families of solutions in term of elliptic functions are derived via Lie‘s approach for the first time. Although no extensive studies are known up to this time a physical background of the considered system cannot exclude in future.

Reflection–transmission quantum Yang–Baxter equations

We explore the reflection-transmission quantum Yang-Baxter equations, arising in factorized scattering theory of integrable models with impurities. The physical origin of these equations is clarified and three general families of solutions are described in detail. Explicit representatives of each family are also displayed. These results allow to establish a direct relationship with the different previous works on the subject and make evident the advantages of the reflection-transmission algebra as an universal approach to integrable systems with impurities.

Evolution of Free Magnetic Fields. I. Neglecting Self-Gravitation

This paper gives the exact solutions, graphical and analytical, for an idealization of a phenomenon described in every textbook of physics: the decay of a magnetic field after its source current has ceased. The problem is solved for the cylindrically symmetric case where a uniform current in a long straight wire is suddenly brought to zero at an initial time $\ensuremath{\tau}=0$. The probable reason why this apparently simple problem has not been solved before is that the mathematical details are complex. The solutions for the electric and magnetic fields which result from the decay of the initial magnetic field are classified according to the four regions of a space-time diagram in which they appear: Regions $A$ (inside wire) and $O$ (outside wire) are outside the light cone of a point on the boundary of the wire. In accord with causality, the magnetic and electric fields retain their initial values (the latter equal to zero) in region $O$; in region $A$ the magnetic field retains its initial value, but the electric field rises linearly with time, since its time derivative replaces the extinguished constant current density. Region $B$ is between the axis and the wave front from the boundary (which started at $\ensuremath{\tau}=0$) after this wave front has passed through the axis and started outward again. Region $C$ is contained between the initially outgoing wave front which starts from the wire boundary and the initially ingoing wave front after it has passed through the axis and started outward again; it is therefore a region which the initial disturbance from the wire boundary has swept over only once instead of twice, in contrast to region $B$. In both regions $B$ and $C$ the analytical expressions for the electric and magnetic fields $\mathcal{E}$ and $\mathcal{B}$ are very complicated except along certain selected lines in space-time, e.g., at the wire axis ($\mathcal{E}={[\ensuremath{\tau}+{({\ensuremath{\tau}}^{2}\ensuremath{-}1)}^{\frac{1}{2}}]}^{\ensuremath{-}1}$, $\mathcal{B}=0$) or on the light cone of a wire boundary point ($\mathcal{E}=\ensuremath{\tau}[1\ensuremath{-}(\frac{2}{\ensuremath{\pi}})invcos(\frac{1}{{\ensuremath{\tau}}^{\frac{1}{2}}})]\ensuremath{-}(\frac{2}{\ensuremath{\pi}}){(\ensuremath{\tau}\ensuremath{-}1)}^{\frac{1}{2}}$, $\mathcal{B}=\frac{1}{2}(\ensuremath{\tau}\ensuremath{-}1)\ensuremath{-}[\frac{\ensuremath{\tau}}{\ensuremath{\pi}{(\ensuremath{\tau}\ensuremath{-}1)}^{\frac{1}{2}}}]{1\ensuremath{-}[\frac{(2\ensuremath{-}\ensuremath{\tau})}{{(\ensuremath{\tau}\ensuremath{-}1)}^{\frac{1}{2}}}]invsin{[\frac{(\ensuremath{\tau}\ensuremath{-}1)}{\ensuremath{\tau}}]}^{\frac{1}{2}}}$). Quite apart from the specific nature of this problem, the investigations in this and the following papers illustrate a practical method for finding realistic solutions to the Einstein-Maxwell time-dependent equations. These solutions correspond to special cases---worked out explicitly, and in one case independently---of a general family of solutions of the time-dependent electrogravitational equations for cyclindrical symmetry. This general family of solutions has been discussed by Melvin and stachel.