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In this paper, we give a proof of the existence of stationary dark soliton solutions or heteroclinic orbits of nonlinear equations of Schr\"odinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of…
We study the quantum mechanics of the derivative nonlinear Schrodinger equation which has appeared in many areas of physics and is known to be classically integrable. We find that the N-body quantum problem is exactly solvable with both…
The dynamics of two-component solitons is studied, analytically and numerically, in the framework of a system of coupled extended nonlinear Schr\"odinger equations, which incorporate the cross-phase modulation,…
An integrable extension of the well known nonlinear Schroedinger (NLS) equation to a higher space-dimension, recently proposed by us, is investigated, exploring its various important aspects. Focusing on the idea of construction its…
The existence of ``dispersion-managed solitons'', i.e., stable pulsating solitary-wave solutions to the nonlinear Schr\"{o}dinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our…
We study the relations between solitons of nonlinear Schr\"{o}dinger equation described systems and eigen-states of linear Schr\"{o}dinger equation with some quantum wells. Many different non-degenerated solitons are re-derived from the…
We study the relationship between the classical Hamilton flow and the quantum Schr\"odinger evolution where the Hamiltonian is a degree-2 complex-valued polynomial. When the flow obeys a strict positivity condition equivalent to compactness…
Evolution algebras are a special class of non-associative algebras exhibiting connections with different fields of Mathematics. Hilbert evolution algebras generalize the concept through a framework of Hilbert spaces. This allows to deal…
In this paper, we consider a general form of nonlinear Schr\"{o}dinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schr\"{o}dinger equation is identified by…
We consider the Schr\"odinger equation with a (matrix) Hamiltonian given by a second order difference operator with nonconstant growing coefficients, on the half one dimensional lattice. This operator appeared first naturally in the…
The Boltzmann type kinetic equation for solitons in Nonlinear Schr\"{o}dinger equation has been constructed on the base of analysis of two soliton collision. Possible applications for Langmuir solitons in plasma and solitons in optic fiber…
The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to algebraic curves.
The paper presents soliton-breather models of particles tunneling on the example of Klein-Gordon and Schrodinger equation nonlinear breathers. It is shown that in this case the non-linearity registration should lead to spatial restrictions…
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the Crocco transformation, with a first-order partial derivative taken as a new independent variable and a second-order…
We investigated the evolute of a space curve with singular points. As smooth curves with singular points, we apply the theory of framed curves. However, the involute corresponding to the evolute in the sense of the locus of the centre of…
Three types of nonlinear Schrodinger models with multiple length scales are considered. It is shown that the length-scale competition universally results into arising of new localized stationary states. Multistability phenomena with a…
We explore and clarify the connections between two different forms of the renormalisation group equations describing the quantum evolution of hadronic structure functions at small $x$. This connection is established via a Langevin…
We study deterministic Loewner evolutions on the complex plane driven by complex-valued functions. This model can be viewed as a generalization of real-driven Loewner evolutions in the upper half-plane, or as the deterministic analogue of…
Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type ("acceleration equals forces") which determine the motion of points in the complex plane. These…
We prove that the solution map for a family of non-linear transport equations in $\mathbb{R}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree…