Single Hump Research Articles

Formation of a Large-Scale Structure in a Nonlinear Dissipative System

Dynamic evolution of a small-scale structure to a large-scale structure is studied by a nonlinear dissipative LC ladder circuit. The circuit is described by a Burgers equation in the lowest order. We confirmed that the observed propagation of a shock wave and the confluence of shock waves are well described by the theory. A single hump wave is used to model a small-scale structure. When two single-hump waves with different amplitudes are applied to the circuit, they evolve into a large-scale structure in the course of propagation. Such a formation of a large-scale structure is unique in a nonlinear dissipative system, while the nonlinear dispersive system is known to preserve the nature of small-scale structures even after their interaction as known for solitons.

Composite multihump vector solitons carrying topological charge

We propose composite solitons carrying topological charge: multicomponent two dimensional [ (2+1)D] vector (Manakov-like) solitons for which at least one component carries topological charge. These multimode solitons can have a single hump or exhibit a multihump structure. The "spin" carried by these multimode composite solitons suggests 3D soliton interactions in which the particlelike behavior includes spin, in addition to effective mass, linear, and angular momenta.

Use of an extremal functional in solving for an unknown bottom surface given a free surface profile

In this paper the free surface fluid flow is induced by an obstacle on the bottom of the channel whose exact shape and location are unknown a priori. The inviscid fluid flow is assumed to be steady, incompressible and irrotational under the influence of gravity. A boundary integral technique, which is based on the combination of the boundary integral method (BIM), the variational principle technique (VPT) and the application of a minimization technique (MT) is developed. Two minimization techniques, namely the extremal pressure method (EPM) and the extremal energy method (EEM), are extensively used in identifying the unknown bottom surfaces. To illustrate these techniques the free surface profile to be applied in the inverse analysis has been generated following a direct formulation when the solid bottom boundary is monotonically decreasing, monotonically increasing, when it possesses a single hump and when it possesses a single depression. It is found that in all four cases both extremum techniques can be very accurately used to identify the shape and position of the solid obstacle.

Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation.

We study an integrable discretization of the nonlinear Schrödinger equation (NLS) under the effects of damping and periodic driving, from the point of view of spatially localized solutions oscillating in time with the driver's frequency. We locate the equilibrium states of the discretized (DNLS) system in the plane of its dissipation gamma and forcing amplitude H parameters and use a shooting algorithm to construct the desired solutions psi(n)(t)=phi(n) exp(it) as homoclinic orbits of a four-dimensional symplectic map in the complex phi(n),phi(n+1) space, for -infinity<n<infinity. We derive, in the gamma=0 case, closed form expressions for two fundamental such solutions having a single hump in n, psi(n)+, and psi(n)-, and determine analytically their threshold of existence in the (gamma,H) plane using Mel'nikov's theory. Then, we demonstrate numerically that above this threshold a remarkable variety of multihump structures appear, whose complexity in terms of their spatial extrema grows with increasing H. All these solutions are numerically found to be unstable in time, except for psi(n)-, which is seen to be stable over a certain region in the (gamma,H) plane. In the continuum limit our results are in close agreement with recent studies on the NLS equation. From a more general perspective, we view these DNLS multihump solutions as homoclinic orbits of a higher-dimensional map thereby providing a possible mechanism for explaining the occurrence of similar structures called discrete (multi-) breathers found in a wide variety of one-dimensional nonlinear lattices.

A nonlinear dynamo wave riding on a spatially varying background

A systematic asymptotic investigation of a pair of coupled nonlinear one–dimensional amplitude equations, which provide a simplified model of solar and stellar magnetic activity cycles, is presented. Specifically, an αΩ –dynamo in a thin shell of small gap–to–radius ratio 𝛆 (≪ 1) is considered, in which the Ω –effect (the differential rotation) is prescribed but the α –effect is quenched by the finite–amplitude magnetic field. The unquenched system is characterized by a latitudinally θ –dependent dynamo number D , with a symmetric single–hump profile, which vanishes at both the pole, θ = π /2, and the equator, θ = 0, and has a maximum, D , at mid–latitude, θ M = π /4. The shape D ( θ )/ D is fixed, so that there is only a single driving parameter D . At onset of global instability, D = D L( e ): = DT + O ( e ), a travelling wave, of frequency ω = ω L( e ): = ω T+ O ( e ) and wavelength O ( e ), is localized at a low latitude θ PT( θ M. In that regime, Meunier and co–workers showed that the O (1) quantities θ F – θ M and ( ω – ω T)/ e 2/3 increase together in concert with D – D T. By analysing the detailed structure of the front of width O ( e ), we obtain ω correct to the higher order O ( e ) and show improved agreement with numerical integrations performed by Meunier and co–workers of the complete governing equations at finite e .

The Interaction of Shocks with Dispersive Waves II. Incompressible‐Integrable Limit

This is the second in a two‐part series of articles in which we analyze a system similar in structure to the well‐known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock‐dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading‐order shock equations and the leading‐order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading‐order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long‐time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast‐time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading‐order shock equations.

High-energy neutron scattering from heavy-fermion systems Ce(In 1− xSn x) 3 and Ce(Pt 1− xRh x) 2

We report neutron-scattering measurements on the heavy-Fermion systems Ce(In 1− x Sn x ) 3 and Ce(Pt 1− x Rh x ) 2. For x = 0, the pure compounds CeIn 3 and CePt 2 order antiferromagnetically at 10.2 and 1.7 K, respectively. Above these temperatures, in the paramagnetic phase, they show a quasi-elastic central ‘peak’ as well as a crystal-field excitation Γ 7 → Γ 8. As x increases the crystal-field excitation becomes ‘quenched’ in both systems and the ground-state ( 2 F 5 2 ) response degenerates into a single broad Lorentzian hump centered on a characteristic energy which increases progressively to a value of ∼ 35 meV for CeSn 3 and as high as ∼ 115 meV for CeRh 2. In the series of compounds Ce(In 1− x Sn x ) 3 the spin-orbit excitation 2 F 5 2 → 2 F 7 2 is found to shift progressively to higher energies with increasing x. The increase with x above the free-ion value of ∼ 273 meV is at a rate roughly twice the characteristic energy of the ground manifold. This implies that the characteristic energy of the 2 F 7 2 state is roughly twice that for the 2 F 5 2 state. Similar enhancements of the spin-orbit excitations in the system Ce(Pt 1− x Rh x ) 2 are expected.

Shape Evolution of Electrodeposited Copper Bumps with High Peclet Numbers

We report the shape evolution of initial copper bumps at Peclet numbers higher than a hundred. The role of vortices and of penetration flow within the cavity was discussed with numerical fluid dynamics computation to obtain a bump with a single hump at the center. The current distributions, or flux profiles, were calculated at the diffusion controlled overpotential and were compared with the electrodeposited bump shapes. For the 100 μm cavity width, the vortices increase at the upstream corners with Peclet numbers 1410 and 7311. The vortices are the local resistance of mass transfer to the cathode. These vortices cause the hollows in flux profiles at the upstream corner with these Peclet numbers. The penetration flow collides with the photoresist sidewall and the vortices decrease at downstream corners. These decreased vortices cause the increase in flux profile at downstream corners. For a 30 μm cavity width a single large vortex forms for the higher Peclet number 44,500 and a single hump in flux is achieved.

Dipole Density Solitons and Solitary Dipole Vortices in an Inhomogeneous Space Plasma.

A new type of density soliton, a dipole density soliton, as well as single dip and hump density solitons, were discovered in recent satellite observations of space plasmas. Moreover, these three kinds of density solitons are all associated with similar local electromagnetic fluctuations with Alfv\'en characteristics. This indicates that they originate from the same physical mechanism. We propose that the solitary plasma dipole vortex model can consistently account for these three kinds of density solitons, with the differences in their appearances attributed to the differences in the positions and directions at which the satellite crosses the solitary dipole vortex.

A linear filter approximation to the hammer/string interaction for use in a commuted synthesis piano model

In commuted synthesis of string instruments, the soundboard/body resonator is commuted to the excitation point and replaced by its own impulse response [Smith and Van Duyne, elsewhere in this session]. Hence, the highly nonlinear hammer/string interaction must be replaced by a commutable linear filter. Using the wave digital hammer computational model of the piano hammer [ 3300(A) (1994)], it was observed that the force pulse of a hammer striking an infinite string was qualitatively similar to the impulse response of a second-order filter with two real poles. Hence, good second- and higher-order filter designs based on physical data were possible. However, multiple humps may appear in the hammer force pulse on a terminated string due to returning string waves. It was observed that the magnitude spectra of the single hump spectrum and the multiple hump spectrum were similar in bandwidth, differing only in a slight ringing in the lower spectrum due to the lowpassed combing effect of the returning string waves. Therefore, an equalization filter was designed to summarize this combing effect by fitting a bank of parallel second-order sections to the complex ratio spectrum. Excellent linear piano hammer simulations were produced.

Using the complex WKB method for studying the evolution of initial pulses obeying the nonlinear Schr�dinger equation

Using the complex WKB method, we found an asymptotic solution of the associated Zakharov-Shabat problem in the limit of a small coefficient h → 0 multiplying the derivative of the potential that has a single hump. The obtained formulas can be used in describing the evolution of optical pulses of such shape obeying the nonlinear Schrodinger equation (NSE). Several examples are considered.

Intensity dependent polarization in a semiconductor multiple quantum-well amplifier

It is shown numerically that the polarization evolution in a semiconductor multiple-quantum-well amplifier is intensity dependent for an input light partially coupled in both polarization modes because of the anisotropy of gain saturation. This phenomenon can be used for nonlinear polarization switching. By placing a polarization beam splitter at the output end, self-polarization switching is obtained. In the case of pulsed inputs, time-dependent polarization is observed, leading to pulse breakup and a poorer switching contrast in nonlinear switching. When the width of the input pulse is much larger than the carrier lifetime, the time-dependent polarization distribution is symmetric, and a strong single hump as well as weak two-hump pulses are observed at the two output ports, respectively.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Influence of stochastic noise on self-organized patterns in a fluid-mosaic model of membranes

A model combining the Smoluchowski equation for the average density of proteins with Kelvin's cable equation modeling diffusion and electrical potential in a membrane, respectively, is considered. The resulting set of nonlinear diffusion equations, which exhibit self-organized stationary patterns, are examined by numerical computations when stochastic noise is introduced. The final stationary patterns have the lowest possible spatial frequencies corresponding to the presence of essentially a single hump. The effect of noise is to enhance the pattern selection.

The development of travelling waves in a simple isothermal chemical system - I. Quadratic autocatalysis with linear decay

The possibility of travelling reaction–diffusion waves developing in the chemical system governed by the quadratic autocatalytic or branching reaction A + B → 2B (rate k 1 ab ) coupled with the decay or termination step B → C (rate k 2 b ) is examined. Two simple solutions are obtained first, namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the reactant B. Both of these indicate that the criterion for the existence of a travelling wave is that k 2 &lt; k 1 a 0 , where a 0 is the initial concentration of reactant A. The equations governing the fully developed travelling waves are then discussed and it is shown that these possess a solution only if this criterion is satisfied, i. e. only if k = k 2 / k 1 a 0 &lt; 1. Further properties of these waves are also established and, in particular, it is shown that the concentration of A increases monotonically from its fully reacted state at the rear of the wave to its unreacted state at the front, while the concentration of B has a single hump form. Numerical solutions of the full initial value problem are then obtained and these do confirm that travelling waves are possible only if k &lt; 1 and suggest that, when this condition holds, these waves travel with the uniform speed v 0 = 2√ (1 – k ). This last result is established by a large time analysis of the full initial value problem that reveals that ahead of the reaction–diffusion front is a very weak diffusion-controlled region into which an exponentially small amount of B must diffuse before the reaction can be initiated. Finally, the behaviour of the travelling waves in the two asymptotic limits k → 0 and k → 1 are treated. In the first case the solution approaches that for the previously discussed k = 0 case on the length scale associated with the reaction–diffusion front, with the difference being seen on a much longer, O ( k –1 ), length scale. In the latter case we find that the concentration of A is 1 + O (1 – k ) and that of B is O ((1 – k ) 2 ), with the thickness of the reaction–diffusion front being of O ((1 – k ) ½ ).

Generalized Burgers equations and Euler–Painlevé transcendents. III

It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986); P. L. Sachdev and K. R. C. Nair, ibid. 28, 977 (1987)] that the Euler–Painlevé equations y(d2y/dη2)+a(dy/dη)2 +f(η)y(dy/dη)+g(η)y2+b(dy/dη) +c=0 represent generalized Burgers equations (GBE’s) in the same way as Painlevé equations represent the Korteweg–de Vries type of equations. The earlier studies were carried out in the context of GBE’s with damping and those with spherical and cylindrical symmetry. In the present paper, GBE’s with variable coefficients of viscosity and those with inhomogeneous terms are considered for their possible connection to Euler–Painlevé equations. It is found that the Euler–Painlevé equation, which represents the GBE ut+uβux=(δ/2)g(t)uxx, g(t)=(1+t)n, β&amp;gt;0, has solutions, which either decay or oscillate at η=±∞, only when −1&amp;lt;n&amp;lt;1. The solutions are shocklike when n=1. On the other hand, they oscillate over the whole real line when n=−1. Furthermore, the solutions monotonically decay both at η=+∞ and η=−∞, that is, they have a single hump form if β≥βn=(1−n)/(1+n). For β&amp;lt;βn, the solutions have an oscillatory behavior either at η=+∞ or at η=−∞, or at η=+∞ and η=−∞. For β=βn, there exists a single parameter family of exact single hump solutions, similar to those found for the nonplanar Burgers equations in Paper II. Thus the parametric value β=βn seems to bifurcate the families of solutions, which remain bounded at η=±∞. Other GBE’s considered here are also found to be reducible to Euler–Painlevé equations. The scope of these equations is broadened by relating them to a large number of nonlinear DE’s selected from the compendia of Kamke [Differential Gleichungen : Lösungsmethoden und Lösungen (Akademische Verlagsgesellschaft, Leipzig, 1943)] and Murphy [Ordinary Differential Equations and their Solutions (Van Nostrand, Princeton, NJ, 1960)]. These latter equations arise from a wide range of physical applications and are of some historical interest as well. They are all special cases of a slightly generalized form of the Euler–Painlevé equation.

The cobweb model: Its instability and the onset of chaos

We introduce a fairly general non-linear supply function into the traditional cobweb model under adaptive expectations. We find that the dynamics of the model are driven by a single hump map of the type that occurs in the chaos literature. By applying some recent results of Feigenbaum we are able to show that in its locally unstable region the cobweb model exhibits a regime of period doubling followed by a chaotic regime.

Chaotic vibrations of a magnet near a superconductor

The forced vibration of a small permanent magnet near the surface of a high temperature superconducting ceramic disc is shown to produce period doubling oscillations and chaos. The source of these anomalous vibrations is the nonlinear, hysteretic force between the magnet and the superconductor. These forces are believed to be related to flux pinning and flux dragging effects in the superconductor in the type II state. A return map, based on the displacement of the magnet, reveals a single hump polynomial function which under iteration exhibits a bifurcation structure similar to the experimental results.

Solitons in Newtonian Gravity and Non-Linear Pancake Evolution

It is shown that the solutions for the non-linear equations governing the motion of a stratified self-gravitating isothermal fluid are generated by a sine-Gordon equation (SG) which has well-known soliton solutions. Starting with two solutions of the SG-equation which are related via an auto-Baecklund-transformation, the physical quantities, density, velocity and gravitational potential are expressed in a parametric form in terms of characteristic coordinates. The zero-solution and the one-soliton solution of SG generate a single hump in the density moving with constant velocity. The typical non-linear feature is the relation between amplitude A and width (y = square of the speed of sound). The 1-soliton together with the 2-sollton solution of SG produces two moving humps in the density which come to rest asymptotically. As a result of the interaction the ratio amplitude/width is different from the corresponding ratio of the single soliton. Due to a certain exchange symmetry in the transformation procedure relating the SG to the gravity system it is possible to generate another solution out of the 2-soliton. Which represents a single hump moving on a homogeneous background. The remarkable property is that the density becomes infinite in a finite time depending on the amplitude. The “breather”-solutlon of SG produces a density distribution that represents two pulses colliding onto a central hump at rest which are separated from each other by a discontinuity in the density and the velocity field. These solutions may have applications to the non-linear evolution of flat structures arising in the process of galaxy formation.

Generalized Burgers equations and Euler–Painlevé transcendents. II

It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986)] that the Euler–Painlevé equation yy″+ay′2+ f(x)yy′+g(x) y2+by′+c=0 represents the generalized Burgers equations (GBE’s) in the same manner as Painlevé equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE ut+uaux+Ju/2t =(gd/2)uxx (the nonplanar Burgers equation) is considered. It is found that its self-similar form is again governed by the Euler–Painlevé equation. The ranges of the parameter α for which solutions of the connection problem to the self-similar equation exist are obtained numerically and confirmed via some integral relations derived from the ODE’s. Special exact analytic solutions for the nonplanar Burgers equation are also obtained. These generalize the well-known single hump solutions for the Burgers equation to other geometries J=1,2; the nonlinear convection term, however, is not quadratic in these cases. This study fortifies the conjecture regarding the importance of the Euler–Painlevé equation with respect to GBE’s.

Generalized Burgers equations and Euler–Painlevé transcendents. I

Initial-value problems for the generalized Burgers equation (GBE) ut+u βux+λuα =(δ/2)uxx are discussed for the single hump type of initial data—both continuous and discontinuous. The numerical solution is carried to the self-similar ‘‘intermediate asymptotic’’ regime when the solution is given analytically by the self-similar form. The nonlinear (transformed) ordinary differential equations (ODE’s) describing the self-similar form are generalizations of a class discussed by Euler and Painlevé and quoted by Kamke. These ODE’s are new, and it is postulated that they characterize GBE’s in the same manner as the Painlevé equations categorize the Kortweg–de Vries (KdV) type. A connection problem for some related ODE’s satisfying proper asymptotic conditions at x=±∞, is solved. The range of amplitude parameter is found for which the solution of the connection problem exists. The other solutions of the above GBE, which display several interesting features such as peaking, breaking, and a long shelf on the left for negative values of the damping coefficient λ, are also discussed. The results are compared with those holding for the modified KdV equation with damping.