Related papers: Conditional Symmetry and Reductions for the Two-Di…
New local smoothing estimates in Besov spaces adapted to the half-wave group are proved via $\ell^2$-decoupling. We apply these estimates to obtain new well-posedness results for the cubic nonlinear wave equation in two dimensions. The…
We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra…
Symmetries and conservation laws are studied for a generalized Westervelt equation which is a nonlinear partial differential equation modelling the propagation of sound waves in a compressible medium. This nonlinear wave equation is widely…
Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are…
It is shown that regularisation by dimensional reduction is a viable alternative to dimensional regularisation in non-supersymmetric theories.
We analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution. Earlier results on…
In this short note, we derive a system of two nonlocal equations for the water-wave problem following the work of [AFM06]. Specifically, we consider a fluid with a one-dimensional free surface for an irrotational fluid both with, and…
The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph…
We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order.…
The vacuum is expected to exhibit electromagnetic nonlinearity. We demonstrate the properties of nonlinear electromagnetic wave in a two-dimensional rectangular cavity by calculating the nonlinear correction for two classical standing…
Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition.
In this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use…
We study the weak universality of the two-dimensional fractional nonlinear wave equation. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $\Phi_2^4$, we first establish a \emph{sufficient and almost…
We prove decay estimates for solutions to non-isotropic linear systems of wave equations. The defining feature of these estimates is that they depend only on the commutation properties of the system with the scaling vector field. As…
Two-dimensional reductions of the KP-Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the Kadomtsev-Petviashvili equation, are studied and…
A recently developed method has been extended to a nonlocal equation arising in steady water wave propagation in two dimensions. We obtain analyic approximation of steady water wave solution in two dimensions with rigorous error bounds for…
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…
For a system of partial differential equations admitting point, contact, or higher symmetries, the framework of invariant reduction systematically computes how invariant geometric structures, such as conservation laws, presymplectic…
To establish a relation between two approaches to the construction of Riemann k-wave solutions of hydrodynamic-type systems, namely the symmetry reduction method and the generalized method of characteristics.
Using systematic calculations in spinor language, we obtain simple descriptions of the second order symmetry operators for the conformal wave equation, the Dirac-Weyl equation and the Maxwell equation on a curved four dimensional Lorentzian…