Nonlinear Superposition Research Articles

Generalized Darboux transformation and asymptotic analysis on the degenerate dark-bright solitons for a coupled nonlinear Schrödinger system

In this paper, our work is based on a coupled nonlinear Schrödinger system in a two-mode nonlinear fiber. A (N,m)-generalized Darboux transformation is constructed to derive the Nth-order solutions, where the positive integers N and m denote the numbers of iterative times and of distinct spectral parameters, respectively. Based on the Nth-order solutions and the given steps to perform the asymptotic analysis, it is found that a degenerate dark-bright soliton is the nonlinear superposition of several asymptotic dark-bright solitons possessing the same profile. For those asymptotic dark-bright solitons, their velocities are z-dependent except that one of those velocities could become z-independent under certain condition, where z denotes the evolution dimension. Those asymptotic dark-bright solitons are reflected during the interaction. When a degenerate dark-bright soliton interacts with a nondegenerate/degenerate dark-bright soliton, the interaction is elastic, and the asymptotic bound-state dark-bright soliton with z-dependent or z-independent velocity could take place under certain condition. Our study extends the investigation on the degenerate solitons from the bright soliton case for the scalar equations to the dark-bright soliton case for a coupled system.

Coupling-managed criticality in nonlinear dynamics of an integrable exciton-phonon system on a one-dimensional lattice

A one-dimensional nonlinear dynamical system of coupled intra-site excitations and lattice vibrations is studied. The system as a whole is shown to be integrable in the Lax sense and it admits the exact four-component analytical solution demonstrating the pronounced mutual influence between the interacting subsystems in the form of essentially nonlinear superposition of two principally distinct types of traveling waves. The interplay between the coupling strength and the parameter of localization causes the criticality of system's dynamics manifested as the dipole-monopole transition in the spatial distribution of intra-site excitations.

Removing mains interference from the mfERG by applying a post-processing digital notch filter: for the good or the bad?

PurposeIdeally, the multifocal electroretinogram (mfERG) is recorded without noticeable intrusion of mains interference. However, sometimes contamination is difficult to avoid. A post-processing digital notch filter can help to recover the retinal response even in severe cases of mains interference. While a digital filter can be designed to have little to no impact on peak times, filtering out mains interference also removes the retinal signal content of the same frequency, which may result in a change of amplitude. The present study addressed this issue in the standard first order kernel mfERG.MethodsIn 24 recordings from routine exams with no perceivable mains interference, the effects of 50-Hz and 60-Hz non-causal digital notch filters on amplitude and peak time were assessed. Furthermore, the effect of filtering on contaminated traces was demonstrated and simulated mains interference was used to provide an example of nonlinear superposition of retinal signal and mains interference.ResultsmfERG amplitudes were reduced by 0%–15% (median 6%) with the 50-Hz filter and remained virtually unaffected with the 60-Hz filter. Simulations illustrate that spurious high-frequency components can occur in the filtered signal if a strongly contaminated signal is clipped due to a limited input range of the analog-to-digital converter.ConclusionThe application of a 50-Hz digital notch filter to mfERG traces causes a mild amplitude reduction which will not normally affect the clinical interpretation of the data. The situation is even more favorable with a 60-Hz digital notch filter. Caution is necessary if the assumption of linear additivity of retinal signal and mains interference is violated.

Magnetic Flux Density Superposition in Nonlinear Anisotropic Ferromagnetic Material and Resulting Magnetic Barkhausen Noise

Tetrapole probes, also known as double-core probes, which consist of two pairs of orthogonal electromagnets, apply orthogonal magnetic fields to the surface of a ferrous sample and rotate the field by superposition of these fields. Tetrapole probes have the potential to perform rapid characterization of the magnetic anisotropy of ferromagnetic materials. Conventional understanding dictates that magnetic field superposition is linear. However, superposition in ferromagnetic materials is inherently nonlinear. In this work, an equation for the magnetic flux density generated by superposition of orthogonal fields from a tetrapole probe in a ferromagnetic, anisotropic, single net easy-axis material, is derived, along with an equation for magnetic Barkhausen noise root mean square (rms) and energy, for application to materials with flat surfaces. The model approximates the nonlinear superposition as cubic, inline with analogous optics results, and is accurate for low applied fields. Data, which are accurately fit with the model, are shown. The model facilitates useful analyses of measurements made by tetrapole probes and addresses the inherent nonlinearity of magnetic fields applied to ferromagnetic materials.

Three-component coupled nonlinear Schrödinger system in a multimode optical fiber: Darboux transformation induced via a rank-two projection matrix

In this paper, the investigation is on a three-component coupled nonlinear Schrödinger (NLS) system which governs the wave evolution in a multimode optical fiber. We construct a Darboux transformation (DT) induced via a rank-two projection matrix, and then derive an (N,m)-generalized DT and the Nth-order solution, where the positive integers N and m denote the numbers of iterative times and distinct spectral parameters, respectively. Focusing on the Nth-order solution on the nonzero–zero–zero background, we derive two kinds of waves which could not be derived by the DT induced via a rank-one projection matrix, that is, the so-called fundamental nonlinear wave (N=1) and degenerate fundamental nonlinear wave (N=2 and m=1). Via the asymptotic analysis, we find that the fundamental nonlinear wave is the nonlinear superposition of two dark–bright–bright solitons and a breather/Kuznetsov–Ma​ breather/rogue wave; and the degenerate fundamental nonlinear wave is the nonlinear superposition of four dark–bright–bright solitons and two breathers/two Kuznetsov–Ma​ breathers/a second-order rogue wave. Since such phenomena are not admitted for the one-component NLS equation and two-component coupled NLS system, they are more useful to understand the three-component coupled NLS system than what the latter two models admit. For other three-component coupled systems, more phenomena may be expected when the rank of projection matrix used to construct a DT is two rather than one, because our study presents an example.

A super KdV equation of Kupershmidt: Bäcklund transformation, Lax pair and related discrete system

A super extension of the Korteweg-de Vries (KdV) equation proposed by Kupershmidt is studied. Its Bäcklund transformation is explicitly presented and a Lax pair is derived, so the integrability of this system is confirmed. The corresponding nonlinear superposition formula is constructed from the Bäcklund transformation and is interpreted as a fully discrete system, whose integrability is demonstrated by verifying its consistency around the cube (CAC) property and supplying a zero-curvature representation.

Research on damage effect of underwater multipoint synchronous explosion shock waves on air-backed clamped circular plate

An underwater multipoint synchronous explosion can enhance the damage effect of an underwater explosion shock wave on a structure. This study investigates the damage effect of an underwater multipoint synchronous explosion on an air-backed clamped circular plate. First, we derive the theoretical formula for the superposition of underwater explosion shock wave pressure. A numerical simulation was conducted using the ALE (Arbitrary Lagrange-Euler) algorithm to investigate the damage effect of underwater explosion shock wave pressure on a clamped circular plate, and an experiment was conducted in a tank. By changing the conditions (explosive distance, explosive spacing, and number of explosives), the damage effect of an underwater multipoint synchronous explosion on an air-backed clamped circular plate was thoroughly investigated. The results show that the shock wave pressure of an underwater two-point synchronous explosion has a nonlinear superposition effect that is approximately 1.48–2.52 times greater than that of a single-point explosion. As the distances between the explosives and between the explosives and the plate decrease, the damage effect increases. When the total mass of the explosives is constant, the synchronous explosion of eight explosives has the greatest damage effect on the clamped circular plate.

Dynamics of transformed nonlinear waves in the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation I: Transitions mechanisms

We explore the dynamical properties of transformed nonlinear waves (TNWs) for the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation describing the propagation of waves in fluids. The breath-wave solution is first given by the Hirota bilinear method. Different from the (1+1)- or (2+1)-dimensional case, three types of conversion conditions are analytically derived in different spatial coordinates, by which the breath waves can be converted into diverse TNWs, including the M-shaped kink soliton, kink soliton with multi peaks, (quasi-) kink soliton, and (quasi-) periodic wave. In addition, an attractive dynamic mechanism of high-dimensional nonlinear waves is investigated, where the shape-changed evolution of these waves can be observed. Then the gradient relationship of the TNWs is illustrated in terms of the wave number ratio of superposition components. The formation mechanism of TNWs is further analyzed based on the analysis of nonlinear superposition and phase shift. Different from previous result, the wave component for the (3+1)-dimensional BKP shows the kink-shaped profile, instead of the bell-shaped one. The principle of the nonlinear superposition is further used to explicate the essence of oscillation, locality and shape-changed evolution of the TNWs. The lump wave is finally transformed into the line rogue wave (LRW) showing the short-lived property. This indicates that the LRWs could be incorporated into the framework of TNWs in some high-dimensional systems.

On the application of generative adversarial networks for nonlinear modal analysis

Linear modal analysis is a useful and effective tool for the design and analysis of structures. However, a comprehensive basis for nonlinear modal analysis remains to be developed. In the current work, a machine learning scheme is proposed with a view to performing nonlinear modal analysis. The scheme is focussed on defining a one-to-one mapping from a latent ‘modal’ space to the natural coordinate space, whilst also imposing orthogonality of the mode shapes. The mapping is achieved via the use of the recently-developed cycle-consistent generative adversarial network (cycle-GAN) and an assembly of neural networks targeted on maintaining the desired orthogonality. The method is tested on simulated data from structures with cubic nonlinearities and different numbers of degrees of freedom, and also on data from an experimental three-degree-of-freedom set-up with a column-bumper nonlinearity. The results reveal the method’s efficiency in separating the ‘modes’. The method also provides a nonlinear superposition function, which in most cases has very good accuracy.

Solitary waves in magnetostrictive circular rods with temperature effect

A simple nonlinear model is constructed in this paper to study the solitary wave in an infinite circular magnetostrictive rod. Based on the constitutive relations for transversely isotropic magnetostrictive materials, considering the coupling of multiphysics, combined with Hamilton’s principle and Euler equation, the longitudinal wave equation (LWE) of the infinite circular rod is obtained. The nonlinearity considered is geometrically associated with the nonlinear normal strain in the longitudinal rod direction. The transverse Poisson’s effect is included by introducing the effective Poisson’s ratio. Solitary wave solution, non-topological bell-type soliton and singular periodic solutions of the LWE are obtained by the [Formula: see text]-expansion method. By using the reductive perturbation method, we derive the KdV equation, furthermore, the two-solitary solution is obtained. Numerical analysis results show that the increase of the magnetic field intensity or temperature will reduce the solitary wave’s propagation velocity. As the wave velocity ratio increases, the wave amplitude gradually increases; when the coupled physics parameter and the wave velocity ratio are constant, the increase of the dispersion parameter will make the wavelength longer. The dynamic behavior of the two-soliton solution in the magnetostrictive rod exhibits nonlinear superposition and has elastic collision characteristics.

Nonlinear wave transitions and their mechanisms of (2+1)-dimensional Sawada–Kotera equation

The transitions and mechanisms of nonlinear waves in the (2+1)-dimensional Sawada–Kotera (2DSK) equation are studied by means of characteristic line and phase shift analysis, and the dynamic behavior of various nonlinear transformed waves is analyzed. Firstly, we obtain the N-soliton solution based on the Hirota bilinear method, from which the breath wave solution is constructed by changing the parameters into a complex form in pairs, and lump solution is obtained via the long wave limit method. Then the mechanism of the breath wave solution transformation is studied by characteristic line analysis, we present the types of transformed nonlinear waves, including quasi-anti-dark soliton, M-shaped soliton, W-shaped soliton, multi-peak soliton, and quasi-periodic wave soliton, and the distribution diagrams of these nonlinear waves on the (α,β) plane is rendered. We further reveal the gradient properties of the transformed wave. In addition, the transformed wave is decomposed into a solitary wave and a periodic wave component, and the formation mechanism, locality, and oscillation properties of the nonlinear transformed wave are explained through the nonlinear superposition. Furthermore, we demonstrate that the geometric properties of the characteristic lines vary with time essentially resulting in the time-varying properties of nonlinear waves, which have never been found in (1+1)-dimensional systems. Based on high-order nonlinear waves, the state transitions of the mixed solution and the second-order breath wave solution are investigated. We show several collision models of nonlinear waves, and reveal that the phase shift difference between the solitary and the periodic wave component leads to the deformable collision of the transformed wave. Such phase shift is due to time evolution and wave interaction. Finally, the dynamic process of nonlinear wave collision under the combined action of time and collision is presented.

Experimental study on the solitary wave-current interaction and the combined forces on a vertical cylinder

Wave-current interaction affects both wave characteristics and current properties and therefore the combined forces on coastal structures. In the first part of this paper, a series of experiments on the solitary wave-current interaction are conducted to investigate how the wave height and water velocity are affected by the wave-current interaction. Then, a vertical cylinder, widely used as the foundation of coastal structures, is mounted into the wave flume. A series of experiments on the solitary wave-current interaction with the cylinder are carried out to investigate the combined flow-induced forces. The experimental results indicate that: 1) when solitary wave travels with following currents (current travels with the same direction of the solitary wave), wave height decreases with increasing current velocity. The effect of following current on wave height reduction is more significant for waves with smaller original wave height and in deeper water depth. When traveling with opposing currents (current travels with the opposing direction of the solitary wave), wave height does not vary monotonically with current velocity but shows an increase followed by a decrease. 2) When a solitary wave travels with following currents, the water velocity near still water level is equal to the linear superposition of water velocity of wave-alone and current-alone. 3) A nonlinear superposition method is suggested to predict the combined solitary wave-following current induced forces.

Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids

In this article, we consider the (3+1)-dimensional generalized Kadomtsev–Petviashvili (GKP) equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the (oscillating-) W- and M-shaped waves, rational W-shaped waves, multi-peak solitary waves, (quasi-) Bell-shaped and W-shaped waves and (quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.

M -Breather, Lumps, and Soliton Molecules for the 2 + 1 -Dimensional Elliptic Toda Equation

The 2 + 1 -dimensional elliptic Toda equation is a higher dimensional generalization of the Toda lattice and also a discrete version of the Kadomtsev-Petviashvili-1 (KP1) equation. In this paper, we derive the M -breather solution in the determinant form for the 2 + 1 -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the 2 + 1 -dimensional elliptic Toda equation are derived from the breather solutions through the degeneration process. Hybrid solutions composed of two line solitons and one breather/lump are constructed. By introducing the velocity resonance to the N -soliton solution, it is found that the 2 + 1 -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the N -soliton solution, we also demonstrate the interactions between a soliton/breather-soliton molecule and a lump and the interaction between a soliton molecule and a breather. It is interesting to find that the KP1 equation does not possess a line soliton molecule, but its discrete version—the 2 + 1 -dimensional elliptic Toda equation—exhibits line soliton molecules.

Resonant line wave soliton solutions and interaction solutions for (2+1)-dimensional nonlinear wave equation

Based on the N-soliton solutions, the resonant line wave soliton and interaction solutions are derived through some constraints in the (2+1)-dimensional nonlinear wave equation. General resonant line wave soliton solutions are firstly presented and their changing routes are illustrated. Then, several interaction solutions including a nonlinear superposition of resonant line wave soliton with breather wave, a hybrid between resonant line wave soliton and lump wave are constructed via long-wave limit method and module resonant mechanism. The characteristics and properties of these interaction solutions are discussed analytically and graphically. Localized wave and interaction solutions of the nonlinear wave models have a great impact on oceanography and physics. The results may be useful in investigating the physical phenomena in shallow water waves and nonlinear optics.

M-breather, M-lump, breather molecules and their interaction solutions for a (2+1)-dimensional KdV equation

Date-Jimbo-Kashiwara-Miwa (DJKM) equation is an integrable (2+1)- dimensional extension of the KdV equation and describes two-dimensional nonlinear dispersive waves. In this paper, we first derive the M-breather solution in terms of determinants for the DJKM equation applying the nonlinear superposition formula and analyze the dynamical properties of the breathers. The M-lump waves for the DJKM equation are obtained through the full degeneration of the breathers and hybrid solutions composed of line solitons, breathers and lumps are constructed. By using the velocity resonance mechanism, we also show that the DJKM equation possesses some resonant structures for breathers and solitons such as breather molecules and breather-soliton molecules. Furthermore, based on the N-soliton solution, the interactions between breather/breather-soliton molecules and breathers/lumps are investigated.

Nonlinear integrable dynamics of coupled vibrational and intra-site excitations on a regular one-dimensional lattice

The nonlinear integrable system of coupled vibrational and intra-site excitations on a regular one-dimensional lattice is studied analytically in the framework of specially developed Darboux–Bäcklund dressing technique. The obtained four-component solution to the system demonstrates the pronounced mutual influence between the subsystems in the form of essentially nonlinear superposition of two principally distinct types of traveling waves. The solution is shown to be critical against the value of localization parameter, so that in under-critical region the subsystem of intra-site excitations should be treated as an extended monopole whereas in over-critical region – as an extended dipole.

A new class of nonlinear superposition between lump waves and other waves for Kadomtsev–Petviashvili I equation

In most cases, the lump wave will collide with line waves and breather waves in the hybrid solutions for (2 + 1)-dimensional integrable systems. However, this study introduces a new constraint condition to construct a nonlinear superposition in which the lump wave does not collide with other waves forever. In particular, the soliton molecule consisting of a lump wave, a line wave and any number of breather waves is derived for the first time by the continued introduction of velocity resonance. The new constraints mentioned in this paper will break the traditional understanding of the interaction between a lump and other waves. And this method can be extended to other (2 + 1)-dimensional integrable systems.

Random forest-based understanding and predicting of the impacts of anthropogenic nutrient inputs on the water quality of a tropical lagoon

Seawater quality degradation is caused by diverse, non-linearly interacting factors, knowledge of which is essential for understanding and predicting water quality trends. Currently, most water-quality research has been based on certain assumptions to employ linear approaches for solving simplified problems, such as numerical simulations or cumulative impact assessments. To improve the accuracy and ease of prediction, the random forest method has been increasingly employed as a good alternative to traditional prediction methods. In the present study, the random forest method was adopted to construct a model of the water quality response of Xincun Lagoon to anthropogenic nutrient inputs based on a limited amount of sample data, aiming to (a) identify the critical sources of nutrient inputs that affect the meeting of water quality objectives so as to minimize the socioeconomic impact on secondary stakeholders; and (b) predict the impact of a reduction of anthropogenic nutrient inputs on water quality improvement. It can be seen from the results that the intensity of stressors generated by different human activities presents an obvious non-linear superposition pattern, and the random forest method is one of the feasible solutions to this phenomenon; in addition, the impact on the lagoon ecosystem is not directly related to the intensity of the pressure source, for example, coastal aquaculture is more important than shallow sea cage aquaculture. Therefore, the method established in this paper can be used to identify the key pressure sources during the restoration of the lagoon environment, so as to achieve the unity of economy and effectiveness.

The dressing method as non linear superposition in sigma models

We apply the dressing method on the Non Linear Sigma Model (NLSM), which describes the propagation of strings on ℝ × S2, for an arbitrary seed. We obtain a formal solution of the corresponding auxiliary system, which is expressed in terms of the solutions of the NLSM that have the same Pohlmeyer counterpart as the seed. Accordingly, we show that the dressing method can be applied without solving any differential equations. In this context a superposition principle emerges: the dressed solution is expressed as a non-linear superposition of the seed with solutions of the NLSM with the same Pohlmeyer counterpart as the seed.