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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).
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 $n>1$ 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.