Related papers: Group classification of variable coefficient KdV-l…

An exact invariant operator of time-dependent coupled oscillators is derived using the Liouville-von Neumann equation. The unitary relation between this invariant and the invariant of two uncoupled simple harmonic oscillators is…

The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic…

In this paper we consider a generalized Kuramoto-Sivashinsky equation. The equivalence group of the class under consideration has been constructed. This group allows us to perform a comprehensive study and a clear and concise formulation of…

Kinematical invariance groups of the 3d Schr\"odinger equations with position dependent masses (PDM) and arbitrary potentials are classified. It is shown that there exist 94 classes of such equations defined up to the generic equivalence…

We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is…

Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the…

In this paper, we focus on a general class of Schr\"odinger equations that are time-dependent and quadratic in X and P. We transform Schr\"odinger equations in this class, via a class of time-dependent mass equations, to a class of solvable…

We study the classification of group actions on C*-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-algebra. We show that a conjecture of…

We determine the class of finite T_0-spaces allowing for a universal coefficient theorem computing equivariant KK-theory by filtrated K-theory.

Maximal kinematical invariance groups of $2d$ Schr\"odinger equation with a position dependent mass and arbitrary potential are classified. It is demonstrated that there exist seven classes of such equations possessing non-equivalent…

We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx, invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space-time reflection-symmetric first order…

In this paper, we combine Bochner formula, Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $\Delta_pu+\Delta_qu+h(u,|\nabla…

We consider initial-boundary value problems for the KdV equation $u_t + u_x + 6uu_x + u_{xxx} = 0$ on the half-line $x \geq 0$. For a well-posed problem, the initial data $u(x,0)$ as well as one of the three boundary values $\{u(0,t),…

This paper is concerned with a class of partial differential equations, which are the linear combinations, with constant coefficients, of the classical flows of the KdV hierarchy. A boundary value problem with inhomogeneous boundary…

This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the…

Here, we study quantitative homogenization of first-order convex Hamilton-Jacobi equations with $(u/\varepsilon)$-periodic Hamiltonians which typically appear in dislocation dynamics. Firstly, we establish the optimal convergence rate by…

The Korteweg-de Vries equation u_t+uu_x+u_{xxx}=0 is PT symmetric (invariant under space-time reflection). Therefore, it can be generalized and extended into the complex domain in such a way as to preserve the PT symmetry. The result is the…

In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed…

We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k\geq5$ is an integer number and $\mu=\pm1$. In the focusing case ($\mu=1$), we show that if the initial data $u_0$…

In the sub-Riemannian setting of Carnot groups, this work investigates a-priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type $$ \Delta_{\mathbb{G}}^\varphi u \ge b(x)…