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A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.
In the paper, arguments are given as to why the concept of static evaluation has the potential to be a useful extension to Monte Carlo tree search. A new concept of modeling static evaluation through a dynamical system is introduced and strengths and weaknesses are discussed. The general suitability of this approach is demonstrated.
Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa–Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor. 43 (2010) 492001 (20pp)]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete set of first-order integrating factors for the systems in Cauchy–Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).