Related papers: On time evolutions associated with the nonstationa…
We present easily verifiable sufficient conditions of time-independence and commutativity for local and nonlocal symmetries for a large class of homogeneous (1+1)-dimensional evolution systems. In contrast with the majority of known…
Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with…
Polynomial-in-time dependent symmetries are analysed for polynomial-in-time dependent evolution equations. Graded Lie algebras, especially Virasoro algebras, are used to construct nonlinear variable-coefficient evolution equations, both in…
The Painleve-Calogero correspondence is extended to auxiliary linear problems associated with Painleve equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the…
We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax…
We introduce an integrable Hamiltonian system which Lax deforms the Dirac operator D=d+d* on a finite simple graph or compact Riemannian manifold. We show that the nonlinear isospectral deformation always leads to an expansion of the…
We generalize a finite parity-time (${\cal PT}$-) symmetric network of the discrete nonlinear Schr\"odinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well…
We consider the long time semiclassical evolution for the linear Schr\"odinger equation. We show that, in the case of chaotic underlying classical dynamics and for times up to $\hbar^{-2+\epsilon},\ \epsilon>0$, the symbol of a propagated…
We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary evolution, and collecting the basic…
Starting with the general description of a moving curve, we have recently presented a unified formalism to show that three distinct space curve evolutions can be identified with a given integrable equation. Applying this to the nonlinear…
This paper investigates the nonlinear Schr\"{o}dinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
We study in this paper the time evolution of $PT$-symmetric non-Hermitian Hamiltonian consisting of periodically driven $SU(1,1)$ generators. A non-Hermitian invariant operator is adopted to solve the Schr\"{o}dinger equation, since the…
We present the explicit formulae, describing the structure of symmetries and formal symmetries of any scalar (1+1)-dimensional evolution equation. Using these results, the formulae for the leading terms of commutators of two symmetries and…
We consider the Cauchy problem of a dissipative nonlinear Schr\"odinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a…
We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior…
We propose a new approximation-technique to deal with the exact macroscopic integro-differential evolution equations of statistical systems which self-consistently accounts for dissipative effects. Concentrating on one and two point…
We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems…
The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the…
We consider a Cauchy Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the…