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Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several $O(N)$-invariant classes of hyperbolic equations $Utx=f(U,Ut,Ux)$ for an $N$-component vector $U(t,x)$ are considered. In each class we find all scaling-homogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented.

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 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.

Travelling waves and conservation laws are studied
for a wide class of $U(1)$-invariant complex mKdV equations
containing the two known integrable generalizations of
the ordinary (real) mKdV equation.
The main results on travelling waves include deriving
new complex solitary waves and kinks that generalize
the well-known mKdV $\sech$ and $\tanh$ solutions.
The main results on conservation laws consist of explicitly finding
all 1st order conserved densities that yield phase-invariant counterparts of
the well-known mKdV conserved densities for
momentum, energy, and Galilean energy,
and a new conserved density describing
the angular twist of complex kink solutions.

Symmetries and conservation laws are studied for two classes
of physically and analytically interesting radial wave equations
with power nonlinearities in multi-dimensions.
The results consist of two main classifications:
all symmetries of point type and all conservation laws of a general energy-momentum type
are explicitly determined,
including those such as dilations, inversions, similarity energies and conformal energies
that exist only for special powers or dimensions.
In particular, all variational cases (when a Lagrangian formulation exists)
and non-variational cases (when no Lagrangian exists)
for these wave equations are considered.
As main results, the classification yields generalized energies and radial momenta
in certain non-variational cases,
which are shown to arise from a new type of Morawetz dilation identity
that produces conservation laws for each of the two classes of wave equations
in a different way than Noether's theorem.

A symmetry group method is used to obtain exact solutions for
a semilinear radial heat equation in $n&gt;1$ 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.