Standing Wave Solutions Research Articles

Ka-band linearizer structure studies for a compact light source

The CompactLight design for a next-generation x-ray free-electron laser utilizes a C-band injector. This requires that the harmonic system used to linearize the beam's phase space must operate at X-band rf or higher. We investigate the optimum frequency for the harmonic system in the range of frequencies from 12 to 48 GHz. We describe the reasoning behind selecting 36 GHz (Ka-band) as our working harmonic frequency. The full linearizer system design including the power source, pulse compressor, and linearizing structure, along with options, is considered and presented. These designs are compared in terms of rf and beam dynamics performance. Two potential MW-level rf sources are discussed; a multibeam klystron and a gyro-klystron, while a klystron-based upconverter with an X-band driver is briefly discussed as an alternative path if even higher peak powers are needed. To further increase peak power, novel options for pulse compressors at Ka-band are discussed. Traveling and standing wave solutions for the structure are presented.

A thin-film waveguide problem with positive Kerr nonlinearity and its TM standing wave solution

In the previous work [1], we applied the scattering-type TM standing wave solution to a slab waveguide with negative Kerr nonlinearity. In this work, we analyzed a waveguide problem with positive Kerr nonlinearity. In our formalism, fields are expressed through the value of a dielectric function and a constant of non-flow integral of motion. We formulated the necessary boundary condition for both boundaries with no reference to a film thickness. The condition binds a dielectric function to have two equal or two different values at boundaries leading to the existence of symmetrically and asymmetrically fit wave pieces. Families of such standing wave solutions were constructed. The satisfactory boundary condition is implemented when we start to apply a film thickness to family data. The family of symmetrically fit wave pieces turned out to have a relatively complex structure due to two singularities in a dielectric function and three regions had to be analyzed separately. To look for matches within the data we presented two simple criteria that completely define the length-form and the number of standing wave solutions within a film. Afterward, a calculation of fitted solutions is straightforward.

Orbital stability and concentration of standing-wave solutions to a nonlinear Schrödinger system with mass critical exponents

Orbital stability and concentration of standing-wave solutions to a nonlinear Schrödinger system with mass critical exponents

Standing Waves of Reaction-Diffusion Equations on an Unbounded Graph with Two Vertices

We deal with bistable reaction-diffusion equations in a domain of a metric graph with two vertices, that is, a domain of multiple half-lines with two junctions connected by a line segment. We prove that there exist two types of standing waves: standing front waves and unimodal waves, if the line segment is long enough. We also numerically show the exact number of standing wave solutions for a cubic nonlinearity and for a piecewise linear case. The stability and instability of the standing wave solutions are also investigated. Standing waves play a role in blocking the front propagation.

Orbital stability of periodic standing waves for the cubic fractional nonlinear Schrödinger equation

In this paper, the existence and orbital stability of the periodic standing wave solutions for the nonlinear fractional Schrödinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding the oscillation theorem for fractional Hill operators and the Vakhitov-Kolokolov condition, well known for Schrödinger equations. We then perform a numerical approach to generate the periodic standing wave solutions of the fNLS equation by using the Petviashvili's iteration method. We also investigate the Vakhitov-Kolokolov condition numerically which cannot be obtained analytically for some values of the order of the fractional derivative.

Standing wave solution for the generalized Jackiw-Pi model

Abstract We study the existence and nonexistence of the standing wave solution for the generalized Jackiw-Pi model by using variational method. Depending on interaction strength λ \lambda , we have three different situations. The existence and nonexistence of the standing wave solution correspond to 1 < λ 1\lt \lambda and 0 < λ < 1 0\lt \lambda \lt 1 , respectively. We have the explicit solution of self-dual equation for the borderline λ = 1 \lambda =1 .

Exploring Structural Flexibility and Stability of α-Synuclein by the Landau-Ginzburg-Wilson Approach.

α-Synuclein (αS) is the principal protein component of the Lewy body and Lewy neurite deposits that are found in the brains of the victims of one of the most prevalent neurodegenerative disorders, Parkinson's disease. αS can be qualified as a chameleon protein because of the large number of different conformations that it is able to adopt: it is disordered under physiological conditions in solution, in equilibrium with a minor α-helical tetrameric form in the cytoplasm, and is α-helical when bound to a cell membrane. Also, in vitro, αS forms polymorphic amyloid fibrils with unique arrangements of cross-β-sheet motifs. Therefore, it is of interest to elucidate the origins of the structural flexibility of αS and what makes αS stable in different conformations. We address these questions here by analyzing the experimental structures of the micelle-bound, tetrameric, and fibrillar αS in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger equation. It is illustrated that without molecular dynamics simulations the kinks are capable of identifying the key residues causing structural flexibility of αS. Also, the stability of the experimental structures of αS is investigated by simulating heating/cooling trajectories using the Glauber algorithm. The findings are consistent with experiments.

Standing wave solutions for a generalized quasilinear Schrödinger equation with indefinite potential

This paper is concerned with the standing waves for a generalized quasilinear Schrödinger equation with indefinite potential. By Morse theory, we obtain a nontrivial solution for the equation. In addition, if the nonlinear term is an odd function, we can obtain an unbounded sequence of solutions.

Standing waves for semilinear Schrödinger equations with discontinuous dispersion

We consider here semilinear Schrödinger equations with a non standard dispersion that is discontinuous at $$x=0$$ . We first establish the existence and uniqueness of standing wave solutions for these equations. We then study the orbital stability of these standing wave into a subspace of the energy space that where classical methods as the concentration-compactness method of P.L. Lions can be used.

Spatiotemporal dynamics in a twisted, circular waveguide array

AbstractWe consider the existence and spectral stability of nonlinear discrete localized solutions representing light pulses propagating in a twisted multicore optical fiber. By considering an even number, N, of waveguides, we derive asymptotic expressions for solutions in which the bulk of the light intensity is concentrated as soliton‐like pulses confined to a single waveguide. The leading order terms obtained are in very good agreement with results of numerical computations. Furthermore, as in the model without temporal dispersion, when the twist parameter, ϕ, is given by , these standing waves exhibit optical suppression, in which a single waveguide remains unexcited, to leading order. Spectral computations and numerical evolution experiments suggest that these standing wave solutions are stable for values of the coupling parameter less than a critical value, at which point a spectral instability results from the collision of an internal eigenvalue with the eigenvalues at the origin. This critical value has a maximum when .

On the stability of the compacton waves for the degenerate KdV and NLS models

In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p p . Namely, we show that all waves are spectrally stable for 2 > p ≤ 8 2>p\leq 8 , while a single mode instability occurs for p > 8 p>8 . This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range p > 8 p>8 .

Soliton solutions for a class of Schrödinger equations with a positive quasilinear term and critical growth

AbstractWe consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics $-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$ in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms $h(s)$ with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$ when the parameter $\kappa >0$ is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.

STANDING WAVE SOLUTIONS FOR THE GENERALIZED MODIFIED CHERN-SIMONS-SCHRÖDINGER SYSTEM

In this paper, we study the modified gauged Schrödinger equation under some assumptions on the functions <i>V</i> and <i>f</i>. By using dual approach, Jeanjean’s monotone trick and Mountain Pass Theorem, we obtain the standing wave solutions for the generalized modified Chern-Simons-Schrödinger system.

Parametrizations of the Poisson–Schrödinger equations in spherical symmetry

We consider the asymptotically flat standing wave solutions to the Poisson–Schrödinger system of equations. These equations are also known as the Schrödinger–Newton equations and are the Newtonian limit of the Einstein–Klein–Gordon equations. The asymptotically flat standing wave solutions to the Poisson–Schrödinger equations are known as static states. These solutions can be parametrized using a variety of choices of two continuous parameters and one discrete parameter, each having a useful physical-geometrical interpretation. The values of the discrete variable determines the number of nodes (zeros) in the solution. We use numerical inversion techniques to analyze transformations between various informative choices of parametrization by relating each of them to a standard set of three parameters. Based on our computations, we propose explicit formulas for these relationships. Our computations also show that for the standard choice of continuous variables, the zero-node ground state yields a minimum value of a geometrically natural discrete variable. We give an explicit formula for this minimum value. We use these results to confirm two related observations from previous work by the author and others, and suggest additional applications and approaches to understand these phenomena analytically.

Classification of resonances and pairing effects on n−A scattering within the Hartree-Fock-Bogoliubov framework

We analyze the properties of the scattering solutions obtained as the pole of the S- and K-matrix with the help of the Jost function framework and the Strum-Liouville theory within the Hartree-Fock-Bogoliubov(HFB) framework, and clarify the scattering solutions which can be defined as the physical state. We found that there are three types of the resonances; "{\it shape resonance}", "{\it particle-type}" and "{\it hole-type quasiparticle resonances}", and another two types of solutions are given as the independent S-matrix and K-matrix poles. The shape resonance is formed by the Hartree-Fock(HF) mean field potential, is not affected by the pairing correlation so much. The particle-type and hole-type quasiparticle resonances originate from the particle and hole states by the configuration mixing effect by pairing. All of resonance are represented by the S-matrix pole which has the corresponding K-matrix pole. Two other types of solutions are given by the independent S-matrix and K-matrix poles. These poles are formed by the HF mean field potential. The effect of pairing for the independent S-matrix pole is small, but the one for the independent K-matrix pole has the remarkable effect. The independent K-matrix pole destroys the quasiparticle resonance as it approaches to the resonance by the pairing effect. The wave function of all resonances have the characteristic structure of the metastable property. However, the metastable structure of the wave function of the quasiparticle resonance can be broken by the independent standing wave solution or the Fano effect.

Discrete breathers and discrete oscillating kink solution in the mass-in-mass chain in the state of acoustic vacuum

The present study is concerned with the dynamics of special localized solutions emerging in the mass-in-mass anharmonic oscillatory chain in the state of acoustic vacuum. Each outer element of the chain incorporates an additional, purely nonlinear mass attachment. Using the homogeneity of the system under consideration, we use the separation of time and space and reduce the analysis of standing wave solutions supported by the chain to the analysis of an algebraic system. An analytical study of the latter revealed the distinct types of static discrete breather solutions as well as the discrete oscillating kinks. Along with the analytical description of their spatial wave profiles, we also establish their zones of existence in the space of system parameters. The stability properties of these solutions are assessed through the linear stability analysis (Floquet). All analytical models are supported by the numerical simulations of the full model.

TM Standing Wave Solution of Maxwell Equations in a Self-Defocusing Kerr Media and its Application to a Thin-Film Waveguide

We applied the dielectric function method to solve analytically L-NL-L structure problems with negative Kerr nonlinearity. A damped wave in linear and a periodic standing wave in non-linear media had to be matched at boundaries. We gave a formulation of boundary conditions that did not explicitly include a film thickness. The boundary-value of a dielectric function can be expressed through the constant of non-trivial integral of motion. Using it, one generates a family of matched solutions satisfying boundary conditions. Then arbitrary film thickness can be checked against this family of solutions in search of matches. As a result, all fitted solutions are determined straightforwardly.

Nonlinear Lorentzian-type standing wave solutions of ac-driven sine-Gordon equation

Readily evaluated nonlinear Lorentzian-type standing wave solutions of ac-driven sine-Gordon equation are presented, by applying a nonfeedback mechanism for chaos control. Three distinct species of Möbius-type standing waves are described: (i) periodic breathers, (ii) hyperbolic breathers, and (iii) Jacobian elliptic breathers. Parameter domains are delineated in which these novel waves exist. For instance, it is exhibited that the amplitude of the periodic breather is proportional to the strength of the ac-driver. For the kink-type and periodic-kink-type waves the strength of the forcing term should be always greater than or equal to 4, in order that these waves to be observed physically as fluxons in long Josephson junctions.

Existence and orbital stability of standing waves to a nonlinear Schr\xf6dinger equation with inverse square potential on the half-line

In our work, we establish the existence of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We apply a profile decomposition argument to overcome the difficulty arising from the non-compactness of the setting. We obtain convergent minimizing sequences by comparing the problem to the problem at “infinity” (i.e., the equation without inverse square potential). Finally, we establish orbital stability/instability of the standing wave solution for mass subcritical and supercritical nonlinearities respectively.

Nonlinear Magneto Convection in a Rotating Fluid due to Vertical Magnetic Field and Vertical Axis of Rotation

In the present paper, linear and weakly nonlinear analysis of magnetoconvection in a rotating fluid due to the vertical magnetic field and the vertical axis of rotation are presented. For linear stability analysis, the normal mode analysis is utilized to find the Rayleigh number which is the function of Taylor number, Magnetic Prandtl number, Thermal Prandtl number and Chandrasekhar number. Also, the correlation between the Rayleigh number and wave number is graphically analyzed. The parameter regimes for the existence of pitchfork, Takens-Bogdanov and Hopf bifurcations are reported. Small-amplitude modulation is considered to derive the Newell-Whitehead-Segel equation and using its phase-winding solution, the conditions for the occurrence of Eckhaus and zigzag secondary instabilities are obtained. The system of coupled Landau-Ginzburg equations is derived. The travelling wave and standing wave solutions for the Newell-Whitehead-Segel equation are also presented. For, standing waves and travelling waves, the stability regions are identified.