Stationary Solitons of the Fifth Order KdV-type Equations and their Stabilization

Exact stationary soliton solutions of the fifth order KdV type equation $$u_t +\alpha u^p u_x +\beta u_{3x}+\gamma u_{5x} = 0$$ are obtained for any p ($>0$) in case $\alpha\beta>0$, $D\beta>0$, $\beta\gamma<0$ (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case $p\geq 5$. Various properties of these solutions are discussed. In particular, it is shown that for any p, these solitons are lower and narrower than the corresponding $\gamma = 0$ solitons. Finally, for p = 2 we obtain an exact stationary soliton solution even when $D,\alpha,\beta,\gamma$ are all $>0$ and discuss its various properties.Comment: 8 pages, no figure

Locking the Golden Door and Throwing Away the Key: An Analysis of Asylum during the Years of the Trump Administration

The years of the Trump Administration have certainly been some of the most divisive in modern American political history. One of the largest divides arose from former President Trump’s brazen, “zero tolerance” immigration policies that relentlessly attacked many forms of immigration coming into the United States. Asylum-based immigration, which allows immigrants to come to this country as a safe haven when they are fleeing persecution in their home countries, was one of former President Trump’s main targets. Former President Trump even came dangerously close to eliminating asylum-based immigration with his “Death to Asylum” policy in December of 2020. President Biden has since reversed many of former President Trump’s detrimental asylum policies and enacted executive orders that facilitate asylum-based immigration. While asylum-based immigration has been saved by President Biden (for now), the actions of the Trump Administration have highlighted the issues regarding lack of consistency and over-delegation to the executive branch that plague immigration law to this day. This Note will examine various sources of asylum law, both prior to and during the Trump Administration, and evaluate the constitutionality of asylum policies between 2016 and 2020. Finally, this Note will give four recommendations future administrations can implement in order to provide fairer and more consistent asylum policies that are not so dependent on which President happens to be in power at the time: (1) creating a direct, fair, and inclusive path to citizenship; (2) decreasing ICE’s role in exchange for increasing the EOIR’s presence; (3) changing the focus in creating available facilities to immigrants; and (4) guaranteeing legal representation in immigration proceedings

Scattering and Trapping of Nonlinear Schroedinger Solitons in External Potentials

Soliton motion in some external potentials is studied using the nonlinear Schr\"odinger equation. Solitons are scattered by a potential wall. Solitons propagate almost freely or are trapped in a periodic potential. The critical kinetic energy for reflection and trapping is evaluated approximately with a variational method.Comment: 9 pages, 7 figure

Chaotic behaviour of nonlinear waves and solitons of perturbed Korteweg - de Vries equation

This paper considers properties of nonlinear waves and solitons of Korteweg-de Vries equation in the presence of external perturbation. For time-periodic hamiltonian perturbation the width of the stochastic layer is calculated. The conclusions about chaotic behaviour in long-period waves and solitons are inferred. Obtained theoretical results find experimental confirmation in experiments with the propagation of ion-acoustic waves in plasma.Comment: 7 pages, LaTeX, 2 Postscript figures, submitted to Reports on Mathematical Physic

Nonlinear Schr\"odinger Equation with Spatio-Temporal Perturbations

We investigate the dynamics of solitons of the cubic Nonlinear Schr\"odinger Equation (NLSE) with the following perturbations: non-parametric spatio-temporal driving of the form $f(x,t) = a \exp[i K(t) x]$, damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a Collective-Coordinate-Theory which yields a set of ODEs for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force $f(x)$. The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of $f(x)$. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum $P(t)$ and soliton velocity $V(t)$: This is a parameter representation of a curve $P(V)$ which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.Comment: 21 figure

Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability

We show that the phenomenon of modulational instability in arrays of Bose-Einstein condensates confined to optical lattices gives rise to coherent spatial structures of localized excitations. These excitations represent thin disks in 1D, narrow tubes in 2D, and small hollows in 3D arrays, filled in with condensed atoms of much greater density compared to surrounding array sites. Aspects of the developed pattern depend on the initial distribution function of the condensate over the optical lattice, corresponding to particular points of the Brillouin zone. The long-time behavior of the spatial structures emerging due to modulational instability is characterized by the periodic recurrence to the initial low-density state in a finite optical lattice. We propose a simple way to retain the localized spatial structures with high atomic concentration, which may be of interest for applications. Theoretical model, based on the multiple scale expansion, describes the basic features of the phenomenon. Results of numerical simulations confirm the analytical predictions.Comment: 17 pages, 13 figure

New features of modulational instability of partially coherent light; importance of the incoherence spectrum

It is shown that the properties of the modulational instability of partially coherent waves propagating in a nonlinear Kerr medium depend crucially on the profile of the incoherent field spectrum. Under certain conditions, the incoherence may even enhance, rather than suppress, the instability. In particular, it is found that the range of modulationally unstable wave numbers does not necessarily decrease monotonously with increasing degree of incoherence and that the modulational instability may still exist even when long wavelength perturbations are stable.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let

Singularites in the Bousseneq equation and in the generalized KdV equation

In this paper, two kinds of the exact singular solutions are obtained by the improved homogeneous balance (HB) method and a nonlinear transformation. The two exact solutions show that special singular wave patterns exists in the classical model of some nonlinear wave problems

Shock waves in the dissipative Toda lattice

We consider the propagation of a shock wave (SW) in the damped Toda lattice. The SW is a moving boundary between two semi-infinite lattice domains with different densities. A steadily moving SW may exist if the damping in the lattice is represented by an ``inner'' friction, which is a discrete analog of the second viscosity in hydrodynamics. The problem can be considered analytically in the continuum approximation, and the analysis produces an explicit relation between the SW's velocity and the densities of the two phases. Numerical simulations of the lattice equations of motion demonstrate that a stable SW establishes if the initial velocity is directed towards the less dense phase; in the opposite case, the wave gradually spreads out. The numerically found equilibrium velocity of the SW turns out to be in a very good agreement with the analytical formula even in a strongly discrete case. If the initial velocity is essentially different from the one determined by the densities (but has the correct sign), the velocity does not significantly alter, but instead the SW adjusts itself to the given velocity by sending another SW in the opposite direction.Comment: 10 pages in LaTeX, 5 figures available upon regues

Instability and Evolution of Nonlinearly Interacting Water Waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schroedinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large amplitude freak waves in deep water.Comment: 4 pages, 4 figures, to appear in Physical Review Letter