On the classification of conditionally integrable evolution systems in (1+1) dimensions

We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order $n$ that admit an $N$-shock type solution with $N\leq n+1$. To this end we develop a refinement of the technique from our earlier work (A. Sergyeyev, J. Phys. A: Math. Gen, 35 (2002), 7653--7660), where we completely characterized all (1+1)-dimensional evolution systems \bi{u}_t=\bi{F}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^n\bi{u}/\p x^n) that are conditionally invariant under a given generalized (Lie--B\"acklund) vector field \bi{Q}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^k\bi{u}/\p x^k)\p/\p\bi{u} under the assumption that the system of ODEs \bi{Q}=0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in $t$, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics. Keywords: Exact solutions, nonlinear evolution equations, conditional integrability, generalized symmetries, reduction, generalized conditional symmetries MSC 2000: 35A30, 35G25, 81U15, 35N10, 37K35, 58J70, 58J72, 34A34Comment: 8 pages, LaTeX 2e, now uses hyperre

On separable Schr\"odinger equations

We classify (1+3)-dimensional Schr\"odinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get eleven classes of the electromagnetic vector potentials of the electromagnetic field $A(t, \vec x)=(A_0(t, \vec x)$, $\vec A(t, \vec x))$ providing separability of the corresponding Schr\"odinger equations. It is established, in particular, that the necessary condition for the Schr\"odinger equation to be separable is that the magnetic field must be independent of the spatial variables. Next, we prove that any Schr\"odinger equation admitting variable separation into second-order ordinary differential equations can be reduced to one of the eleven separable Schr\"odinger equations mentioned above and carry out variable separation in the latter. Furthermore, we apply the results obtained for separating variables in the Hamilton-Jacobi equation.Comment: 30 pages, LaTe

On separable Fokker-Planck equations with a constant diagonal diffusion matrix

We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients $B_1(\vec x),B_2(\vec x),B_3(\vec x)$ providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients $\vec B(\vec x)$ must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructedComment: 20 pages, LaTe

Group classification of heat conductivity equations with a nonlinear source

We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are three, seven, twenty eight and twelve inequivalent classes of partial differential equations of the considered type that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any partial differential equation belonging to the class under study and admitting symmetry group of the dimension higher than four is locally equivalent to a linear equation. This classification is compared to existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page

Decay of metastable phases in a model for the catalytic oxidation of CO

We study by kinetic Monte Carlo simulations the dynamic behavior of a Ziff-Gulari-Barshad model with CO desorption for the reaction CO + O $\to$ CO$_2$ on a catalytic surface. Finite-size scaling analysis of the fluctuations and the fourth-order order-parameter cumulant show that below a critical CO desorption rate, the model exhibits a nonequilibrium first-order phase transition between low and high CO coverage phases. We calculate several points on the coexistence curve. We also measure the metastable lifetimes associated with the transition from the low CO coverage phase to the high CO coverage phase, and {\it vice versa}. Our results indicate that the transition process follows a mechanism very similar to the decay of metastable phases associated with {\it equilibrium} first-order phase transitions and can be described by the classic Kolmogorov-Johnson-Mehl-Avrami theory of phase transformation by nucleation and growth. In the present case, the desorption parameter plays the role of temperature, and the distance to the coexistence curve plays the role of an external field or supersaturation. We identify two distinct regimes, depending on whether the system is far from or close to the coexistence curve, in which the statistical properties and the system-size dependence of the lifetimes are different, corresponding to multidroplet or single-droplet decay, respectively. The crossover between the two regimes approaches the coexistence curve logarithmically with system size, analogous to the behavior of the crossover between multidroplet and single-droplet metastable decay near an equilibrium first-order phase transition.Comment: 27 pages, 22 figures, accepted by Physical Review

Conditional Lie-B\"acklund symmetry and reduction of evolution equations.

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-B\"acklund vector field. Such generalization proves to be effective and enables us to construct principally new Ans\"atze reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations $u_t=u_{xx}+F(u,u_x)$ with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions can not be constructed by means of the symmetry reduction procedure.Comment: 12 pages, latex, needs amssymb., to appear in the "Journal of Physics A: Mathematical and General" (1995

Nonlinear Evolution Equations Invariant Under Schroedinger Group in three-dimensional Space-time

A classification of all possible realizations of the Galilei, Galilei-similitude and Schroedinger Lie algebras in three-dimensional space-time in terms of vector fields under the action of the group of local diffeomorphisms of the space \R^3\times\C is presented. Using this result a variety of general second order evolution equations invariant under the corresponding groups are constructed and their physical significance are discussed