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We prove the existence of strong and weak solutions to the semilinear wave equation with coefficients depending both on time and space variables, with continuous nonlinearity satisfying the sign condition. The uniqueness is proven under…
In his 2007 monograph, D. Christodoulou proved a breakthrough result giving a detailed description of the formation of shocks in solutions to the relativistic Euler equations in three spatial dimensions. He assumed that the data have small…
We present full classification of Q-conditional symmetries for the two-dimensional nonlinear wave equation.
We discuss the singularities in the moduli space of string compactifications to six dimensions with $N=1$ supersymmetry. Such singularities arise from either massless particles or non-critical tensionless strings. The points with…
The peculiar intergrability of the Davey-Stewartson equation allows us to find analytically solutions describing the simultaneous formation and interaction of one-dimensional and two-dimensional localized coherent structures. The predicted…
In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2+1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data…
The interaction of both scalar and counter-rotating polarized steady state pulses (SSP) is studied numerically for a medium characterized by nonlinear susceptibilities of the third and the fifth order (a cubic-quintic medium…
We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of…
In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) =…
We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle…
In nonrelativistic quantum mechanics the spontaneous generation of singularities in smooth and finite wave functions, is a well understood phenomenon also occurring for free particles. We use the familiar analogy between the two-dimensional…
I discuss a special class of singularities obtained as a natural 4-dimensional generalization of the conical singularity. Such singularities (called quasiregular) are ruinous for the predictive force of general relativity, so one often…
We study propagation of phase space singularities for a Schr\"odinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. Phase space singularities are measured by the lack of…
A novel strongly nonlinear laminar metamaterial supporting new types of solitary and shock waves with impact energy mitigating capabilities is presented. It consists of steel plates with intermittent polymer toroidal rings acting as…
Resonant mode interactions in weakly nonlinear multi-dimensional lattices and related effects are described. We concentrate on formal description of the phenomenon and consider as examples mode interactions and evolution equations for…
Power induced wave collapse is one of the most fascinating phenomena in optics as it provides extremely high intensities, thus stimulating a range of nonlinear processes. For low power levels, propagation of beams in bulk media is dominated…
We study the linear evolution of small perturbations in self-gravitating fluid systems in two spatial dimensions; we consider both cylindrical and cartesian (i.e., slab) geometries. The treatment is general, but the application is to…
In this manuscript we first give the explicit variational structure of the nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in 2-D. Based on this variational structure, we suggest a null condition which is a kind…
In a recent letter, Carpenter and Guha (2019) compared the neutral modes of a smooth two dimensional shear profile without an inflection point to the modes of its corresponding piecewise-linear profile. The regular mode in the smooth…
In this paper, we study the problem of the nonlinear interaction of impulsive gravitational waves for the Einstein vacuum equations. The problem is studied in the context of a characteristic initial value problem with data given on two null…