274 research outputs found
Wide Localized Solitons in Systems with Time and Space-Modulated Nonlinearities
In this work we apply point canonical transformations to solve some classes
of nonautonomous nonlinear Schr\"{o}dinger equation namely, those which possess
specific cubic and quintic - time and space dependent - nonlinearities. In this
way we generalize some procedures recently published which resort to an ansatz
to the wavefunction and recover a time and space independent nonlinear equation
which can be solved explicitly. The method applied here allow us to find wide
localized (in space) soliton solutions to the nonautonomous nonlinear
Schr\"{o}dinger equation, which were not presented before. We also generalize
the external potential which traps the system and the nonlinearities terms.Comment: 19 pages, 5 figure
Helmholtz bright and boundary solitons
We report, for the first time, exact analytical boundary solitons of a generalized cubic-quintic Non-Linear Helmholtz (NLH) equation. These solutions have a linked-plateau topology that is distinct from conventional dark soliton solutions; their amplitude and intensity distributions are spatially delocalized and connect regions of finite and zero wave-field disturbances (suggesting also the classification as 'edge solitons'). Extensive numerical simulations compare the stability properties of recently-reported Helmholtz bright solitons, for this type of polynomial non-linearity, to those of the new boundary solitons. The latter are found to possess a remarkable stability characteristic, exhibiting robustness against perturbations that would otherwise lead to the destabilizing of their bright-soliton counterpart
Complex plane representations and stationary states in cubic and quintic resonant systems
Weakly nonlinear energy transfer between normal modes of strongly resonant
PDEs is captured by the corresponding effective resonant systems. In a previous
article, we have constructed a large class of such resonant systems (with
specific representatives related to the physics of Bose-Einstein condensates
and Anti-de Sitter spacetime) that admit special analytic solutions and an
extra conserved quantity. Here, we develop and explore a complex plane
representation for these systems modelled on the related cubic Szego and LLL
equations. To demonstrate the power of this representation, we use it to give
simple closed form expressions for families of stationary states bifurcating
from all individual modes. The conservation laws, the complex plane
representation and the stationary states admit furthermore a natural
generalization from cubic to quintic nonlinearity. We demonstrate how two
concrete quintic PDEs of mathematical physics fit into this framework, and thus
directly benefit from the analytic structures we present: the quintic nonlinear
Schroedinger equation in a one-dimensional harmonic trap, studied in relation
to Bose-Einstein condensates, and the quintic conformally invariant wave
equation on a two-sphere, which is of interest for AdS/CFT-correspondence.Comment: v2: version accepted for publicatio
Special geometry on the 101 dimesional moduli space of the quintic threefold
A new method for explicit computation of the CY moduli space metric was
proposed by the authors recently. The method makes use of the connection of the
moduli space with a certain Frobenius algebra. Here we clarify this approach
and demonstrate its efficiency by computing the Special geometry of the
101-dimensional moduli space of the quintic threefold around the orbifold
point.Comment: We made exposition more clear, in particular we explained how to
generalize our idea
Discrete Nonlinear Schrodinger Equations with arbitrarily high order nonlinearities
A class of discrete nonlinear Schrodinger equations with arbitrarily high
order nonlinearities is introduced. These equations are derived from the same
Hamiltonian using different Poisson brackets and include as particular cases
the saturable discrete nonlinear Schrodinger equation and the Ablowitz-Ladik
equation. As a common property, these equations possess three kinds of exact
analytical stationary solutions for which the Peierls-Nabarro barrier is zero.
Several properties of these solutions, including stability, discrete breathers
and moving solutions, are investigated
On a Class of Spatial Discretizations of Equations of the Nonlinear Schrodinger Type
We demonstrate the systematic derivation of a class of discretizations of
nonlinear Schr{\"o}dinger (NLS) equations for general polynomial nonlinearity
whose stationary solutions can be found from a reduced two-point algebraic
condition. We then focus on the cubic problem and illustrate how our class of
models compares with the well-known discretizations such as the standard
discrete NLS equation, or the integrable variant thereof. We also discuss the
conservation laws of the derived generalizations of the cubic case, such as the
lattice momentum or mass and the connection with their corresponding continuum
siblings.Comment: Submitted for publication in a journal on October 14, 200
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