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Institute
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide.
The algorithmic aspect of our work provides a method to compute rational solutions of single polynomial equations that are typically large with 10^2 ... 10^5 terms and that are heavily underdetermined.
It was possible to obtain this functionality by adding a number of new modules for a new type of splitting of equations to the existing package CRACK that is normally used to solve polynomial algebraic and differential systems of equations.
Recently, Holm and Ivanov, proposed and studied a class of multi-component
generalisations 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,
{\it J. Phys A: Math. Theor} {\bf 43}, 492001 (20pp), 2010]. 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 sent 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).
The paper reports on a computer algebra program {\sc LSSS} (Linear Selective
Systems Solver) for solving linear algebraic systems with rational
coefficients. The program is especially efficient for very large sparse
systems that have a solution in which many variables take the value
zero. The program is applied to the symmetry investigation of a non-abelian
Laurent ODE introduced recently by M.\ Kontsevich. The computed symmetries
confirmed that a Lax pair found for this system earlier generates all first
integrals of degree at least up to 14.
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.
We consider systems of ODEs with the right hand side being Laurent
polynomials in several non-commutative unknowns. In particular,
these unknowns could be matrices of arbitrary size. An important
example of such a system was proposed by M. Kontsevich. We prove the
integrability of the Kontsevich system by finding a Lax pair,
corresponding first integrals and commuting flows. We also provide
a pre-Hamiltonian operator which maps gradients of integrals for
the Kontsevich system to symmetries.
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.
A solution generating technique is developed for $D=5$ minimal supergravity with two commuting Killing vectors based on the $G_2$ U-duality arising in the reduction of the theory to three dimensions. The target space of the corresponding 3-dimensional sigma-model is the coset $G_{2(2)}/(SL(2,R)\times SL(2,R))$. Its isometries constitute the set of solution generating symmetries. These include two electric and two magnetic Harrison transformations with the corresponding two pairs of gauge transformations, three $SL(2,R) \; S$-duality transformations, and the three gravitational scale, gauge and Ehlers transformations (altogether 14). We construct a representation of the coset in terms of $7\times 7$ matrices realizing the automorphisms of split octonions. Generating a new solution amounts to transforming the coset matrices by one-parametric subgroups of $G_{2(2)}$ and subsequently solving the dualization equations. Using this formalism we derive a new charged black ring solution with two independent parameters of rotation.