scholarly journals Finite dimensional varieties on hypergroups

2021 ◽  
Author(s):  
László Székelyhidi ◽  
Żywilla Fechner
Keyword(s):  
Partial Differential Equations ◽  
Positive Integer ◽  
Differential Equations ◽  
Gelfand Pair ◽  
Partial Differential ◽  
Finite Dimensional ◽  
Affine Groups

AbstractLet X be a hypergroup, K its compact subhypergroup and assume that (X, K) is a Gelfand pair. Connections between finite dimensional varieties and K-polynomials on X are discussed. It is shown that a K-variety on X is finite dimensional if and only if it is spanned by finitely many K-monomials. Next, finite dimensional varieties on affine groups over $${\mathbb {R}}^d$$ R d , where d is a positive integer are discussed. A complete description of those varieties using partial differential equations is given.

scholarly journals $ L^p $-exact controllability of partial differential equations with nonlocal terms

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Luisa Malaguti ◽  
Stefania Perrotta ◽  
Valentina Taddei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Controllability ◽  
Partial Differential ◽  
State Spaces ◽  
Strongly Continuous Semigroups ◽  
Infinite Dimensional ◽  
Semilinear Equations ◽  
Finite Dimensional ◽  
Nonlocal Terms

<p style='text-indent:20px;'>The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces, <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p&lt;\infty $\end{document}</tex-math></inline-formula>. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.</p>

scholarly journals Variational principles for stochastic soliton dynamics

2016 ◽  
Vol 472 (2187) ◽  
pp. 20150827 ◽  
Author(s):  
Darryl D. Holm ◽  
Tomasz M. Tyranowski
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Stochastic Modelling ◽  
Partial Differential ◽  
Stochastic Perturbations ◽  
Finite Dimensional ◽  
Stochastically Perturbed

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa–Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations. In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler–Poincaré structure of the CH equation (parametric stochastic deformations, P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary partial differential equation and the sensitivity of the resulting solutions to the choices made in stochastic modelling.

scholarly journals Dynamics of media with stationary parameters and telegraph equations

2021 ◽  
Vol 2091 (1) ◽  
pp. 012069
Author(s):  
A G Kushner ◽  
E N Kushner
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Natural Extension ◽  
Mathematical Physics ◽  
Telegraph Equation ◽  
Partial Differential ◽  
Finite Dimensional ◽  
Telegraph Equations ◽  
Equations Of Mathematical Physics

Abstract The paper proposes an approach for constructing exact solutions of differential equations of mathematical physics, in particular, the telegraph equation. The method is based on the theory of finite-dimensional dynamics of systems of evolutionary differential equations. This theory is a natural extension of the theory of dynamical systems to partial differential equations. It allows one to construct exact solutions of partial differential equations even in the case when equations do not have symmetry algebras sufficient for integration.

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