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Institute
An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists. Previous methods yielded such conditions from admitted point or contact symmetries of the nonlinear PDE. Through examples, these two linearization approaches are contrasted.
A symmetry group method is used to obtain exact solutions for
a semilinear radial heat equation in $n>1$ dimensions
with a general power nonlinearity.
The method involves an ansatz technique to solve
an equivalent first-order PDE system of similarity variables
given by group foliations of this heat equation,
using its admitted group of scaling symmetries.
This technique yields explicit similarity solutions as well as
other explicit solutions of a more general (non-similarity) form
having interesting analytical behavior connected with blow up and dispersion.
In contrast,
standard similarity reduction of this heat equation gives
a semilinear ODE that cannot be explicitly solved by familiar
integration techniques such as point symmetry reduction or integrating factors.
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 is based on group foliation reduction
and employs 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 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.