We present a new stochastic differential equation model for the spontaneous emission noise and carrier noise in semiconductor lasers. The correlations between these two types of noise have often been neglected in recent studies of the effects of the...
We consider the problem of recovering a spatially-localized cubic nonlinearity in a nonlinear Schrödinger equation in dimensions two and three. We prove that solutions with data given by small-amplitude wave packets accrue a nonlinear phase that det...
We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb{R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of...
The Colebrook equation $ζ$ is implicitly given in respect to the unknown flow friction factor $λ$; $λ=ζ(Re,ε^*,λ)$ which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. Common approach to sol...
We derive all the reductions of the system of two coupled sine-Gordon equations introduced by Konopelchenko and Rogers to ordinary differential equations. All these reductions are degeneracies of a master reduction to an equation found by Chazy "curi...
We find dispersion-free wavepacket solutions to the Klein-Gordon equation, with the only free parameter being the wavepacket velocity $ {\bf v} $. These wavefunctions are eigenvectors of a velocity operator with commuting components which is symmet...
The free Schrodinger equation has constant velocity wavepacket solutions ψ_{\bf v} of the form ψ= f({\bf r} - {\bf v}t) e^{- i m c^2 t / 2}. These solutions are eigenvectors of a momentum operator {\bf \tilde p} which is symmetric in a positive d...
We consider the problem of verifying the existence of $H^1$ ground states of the 1D nonlinear Schrödinger equation for an interface of two periodic structures: $$-u" +V(x)u -λu = Γ(x) |u|^{p-1}u \ {on} \R$$ with $V(x) = V_1(x), Γ(x)=Γ_1(x)$ for...
A de Broglie-Bohm like model of Dirac equation, that leads to the correct Pauli equations for electrons and positrons in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, t...
We propose quite a new method of analyzing the dynamical chiral symmetry breaking in gauge theories. Starting with the non-perturbative renormalization group equation for the Wilsonian fermion potential, we define the weak solution of it in order to...
We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-γ}$ is proved. As a corollary, we obtain a global existence result for...
In the present manuscript, we determine the critical condition for the nonlinearity in a semilinear wave equation with a derivative-type nonlinearity. More precisely, we consider a nonlinear term depending on the time derivative of the solution, whic...
We prove that the non-commutative Kadomtsev-Petviashvili (KP) equation and a `lifted' modified Kadomtsev-Petviashvili (mKP) equation are directly linearisable, and thus integrable in this sense. There are several versions of the non-commutative mKP e...
We will find Green's function for the standard weighted Laplacian and use the corresponding Green's potential to solve Poisson's equation in the unit disc with zero boundary values, in the sense of radial $L^1$-means, for complex Borel measures $μ$...
We study positive bound states for the equation $$- ε^2 Δu + Vu = u^p, \qquad \text{in $\mathbf{R}^N$}, $$ where $ε> 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential. Using purely variational techniq...
The paper continues the analysis started in [Cora-Fioravanti-Vita-25,Fioravanti-24] on the local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold. The model operator is g...
The concept of the PT-symmetry, originating from the quantum field theory, has been intensively investigated in optics, stimulated by the similarity between the Schrödinger equation and the paraxial wave equation that governs the propagation of ligh...
We review Darboux-Crum transformation of Heun's differential equation. By rewriting an integral transformation of Heun's differential equation into a form of elliptic functions, we see that the integral representation is a generalization of Darboux...
Magnetization dynamics is commonly described by the stochastic Landau-Lifshitz-Gilbert (LLG) equation. On picosecond timescales, inertial and open-system extensions of the LLG equation are necessary to interpret recent experiments. We show analytical...
Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equation for moderate values of the delay close to the onset of chaos. In pa...