Localized Breathers Research Articles

On examining the predictive capabilities of two variants of the PINN in validating localized wave solutions in the generalized nonlinear Schrödinger equation

We introduce a novel neural network structure called strongly constrained theory-guided neural network (SCTgNN), to investigate the behaviour of the localized solutions of the generalized nonlinear Schrödinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, the NLS, Hirota, Lakshmanan–Porsezian–Daniel and fifth-order NLS equations. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the physics-informed neural network and theory-guided neural network (TgNN) models, the SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviour of localized waves, including solitons, rogue waves and breathers, within the generalized NLS equation. We then use the SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both the SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviour and provide predictions across a wide variety of situations.

Nonlinear localized modes in a higher-order anisotropic ferromagnetic nanowire with octupole–dipole interaction

Increasing demand for advanced technologies that depends on magnetic phenomena, understanding and controlling the behavior of discrete breather in ferromagnetic nanowires are crucial for enhancing the efficiency and performance of such devices. The presence of octupole–dipole interactions signifies a unique aspect that could potentially influence the stability and localization of breather excitations. Hence, we adopted a multifaceted approach to investigate the Heisenberg anisotropic ferromagnetic nanowire discrete model with the following interactions: bilinear, octupole–dipole, anisotropy and its higher-order terms. The dynamics is governed by a discrete nonlinear Schrödinger equation (DNLS) arrived with the aid of Holstein–Primakoff transformation. This transformation was facilitated by utilizing the Glauber coherent representation of the boson operators. Subsequently, the dynamical equation is incorporated to the Modulational Instability (MI) analysis which is a systematical gateway to explore the breather excitation in the region of instability influenced by the octupole–dipole interaction coupling parameter. Then, we pictorially demonstrated that the octupole–dipole interaction plays a pivotal role in promoting the localization of discrete breather on the surface of the spin lattice sites in the discrete ferromagnetic nanowire. The energy density distribution also implies that the increase in octupole–dipole interaction results in the highly dense breather localization. The result shows that the increment in the octupole–dipole interaction parameter increases the amplitude of the localized breathers. These discrete breathers could hold immense promise for applications in magnetic storage and Spintronic devices, where maintaining stable localized modes is crucial for the device functionality. Our novelty lies in being pioneers in the exploration of a fully discrete model that encompasses higher-order interactions, such as the octupole–dipole interaction. We already have confirmed the existence of instability region on the discrete spin lattice by incorporating the octupole–dipole interaction [T. Pavithra, L. Kavitha, Prabhu and A. Mani, Modulational instability analysis in an isotropic ferromagnetic nanowire with higher order octopole-dipole interaction, in Nonlinear Dynamics and Applications: Proceedings of the ICNDA 2022 (Springer, 2022), p. 1209], we attempting to explore the generation of discrete breathers in a discrete anisotropic ferromagnetic nanowire. This effectively bridges the gap between theoretical understanding and practical implications, paving the way for innovative advancements in magnetic technology.

Localized waves and interaction solutions to an integrable variable coefficient Date-Jimbo-Kashiwara-Miwa equation

This paper initiates an exploration into the exact solutions of the variable coefficient Date-Jimbo-Kashiwara-Miwa equation, first utilizing the Painlevé analysis method to discuss the integrability of the equation. Subsequently, By employing the Hirota bilinear method, N-soliton solutions for the equation are constructed. The application of the Long wave limit method to these N-soliton solutions yields rational and semirational solutions. Various types of localized waves, encompassing solitons, lumps, breather waves, and others, emerge through the careful selection of specific parameters. By analyzing the image of the solutions, the evolution process and its dynamical behavior are studied.

Interactions of certain localized waves for an extended (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics

Currently, there is much interest in the study on the localized waves and their interactions in fluid mechanics. An extended (3+1)-dimensional Kadomtsev-Petviashvili equation is considered here. By means of the Hirota method, we obtain the N-soliton solutions, with N as a positive integer. The higher-order breather solutions are obtained from the N-soliton solutions through the complex conjugated transformations. Making use of the long-wave limit method, we determine the higher-order lump solutions via the N-soliton solutions. Besides, some hybrid solutions are presented. Three kinds of the localized waves, namely, the solitons, breathers and lumps, along with their interactions, are investigated via the above solutions. Amplitudes, shapes and velocities of those localized waves remain invariant after the interactions, which indicate that the interactions are elastic. Fluid-mechanically meaningful, all our results rely on the coefficients in that equation.

Attractive Impurity as a Generator of Wobbling Kinks and Breathers in the $\varphi^{4}$ Model

The $\varphi^{4}$ theory is widely used in many areas of physics, from cosmology and elementary particle physics to biophysics and condensed matter theory. However, in the $\varphi^{4}$ model, there are no spatially localized solutions in the form of breathers. Topological defects, or kinks, in this theory describe stable, solitary wave excitations. In practice, these excitations, as they propagate, necessarily interact with impurities or imperfections in the on-site potential. In this work, with the help of numerical calculations using the method of lines, the interaction of the kink in the $\varphi^{4}$ model with extended impurities is considered. The case of an attractive rectangular impurity is analyzed. It is found that after the kink-impurity interaction, an internal mode with frequency $\sqrt{\frac{3}{2}}$ is excited on the kink and it becomes a wobbling kink. It is shown that with the help of kink-impurity interaction, an extended rectangular attracting impurity, as well as a point impurity, can be used as a generator for excitation of long-lived high-amplitude localized breather waves. The structure of the excited wobbling breather (or wobbler), which consists

Localized waves and their novel interaction solutions for a dimensionally reduced (2 + 1)-dimensional Kudryashov Sinelshchikov equation

Propagation of the pressure waves in a liquid with gas bubbles is an important topic in the field of fluid dynamics and mathematical physics. The Kudryashov-Sinelshchikov equation is one of the models that describe the propagation of nonlinear waves in a bubbly liquid taking into consideration the viscosity of the liquid and the heat transfer. To explain such behaviors, we mainly focus in this study to explain the dynamics of localized waves and their variety of interaction solutions to a dimensionally reduced (2 + 1)-dimensional Kudryashov-Sinelshchikov equation with the aid of the Hirota bilinear method from N-soliton solutions. Four different forms of localized waves, including solitons, lumps, breathers, and rogues, are derived from the aforesaid equation based on the long wave limit approach. In particular, the localized waves can be used to find interaction solutions, which are the single breather or single lump formed by two solitons; interaction between one line soliton and one breather, as well as one line soliton and one lump soliton among the three solitons; interaction of the two-line soliton and one periodic breather, two periodic breathers, periodic breather and one lump soliton from the four solitons. The direction of propagation, phase shifts, shape, energy, and the variety of interaction solutions of localized waves are affected by these parameters. Moreover, analytical and graphical illustrations of these interaction solutions and their propagation properties are shown by the three-dimensional and density plots with the help of Maple 17. These newly discovered solutions in this study can be used to illustrate the interaction phenomenon of localized waves on ocean surfaces.

Second-harmonic generation in the system with fractional diffraction

We construct a family of bright optical solitons composed of fundamental-frequency (FF) and second-harmonic (SH) components in the one-dimensional (planar) waveguide with the quadratic (second-harmonic-generating) nonlinearity and effective fractional diffraction, characterized by the Lévy index α, taking values between 2 and 0.5, which correspond to the non-fractional diffraction and critical collapse, respectively. The existence domain and stability boundary for the solitons are delineated in the space of α, FF-SH mismatch parameter, and propagation constant. The stability boundary is tantamount to that predicted by the Vakhitov–Kolokolov criterion, while unstable solitons spontaneously evolve into localized breathers. A sufficiently weak transverse kick applied to the stable solitons excite small internal vibrations in the stable solitons, without setting them in motion. A stronger kick makes the solitons’ trajectories tilted, simultaneously destabilizing the solitons.

Existence of exponentially and superexponentially spatially localized breather solutions for nonlinear klein–gordon lattices in ℤd, d ≥ 1

AbstractWe prove the existence of exponentially and superexponentially localized breather solutions for discrete nonlinear Klein–Gordon systems. Our approach considers $d$-dimensional infinite lattice models with general on-site potentials and interaction potentials being bounded by an arbitrary power law, as well as, systems with purely anharmonic forces, cases which are much less studied particularly in a higher-dimensional set-up. The existence problem is formulated in terms of a fixed-point equation considered in weighted sequence spaces, which is solved by means of Schauder's Fixed-Point Theorem. The proofs provide energy bounds for the solutions depending on the lattice parameters and its dimension under physically relevant non-resonance conditions.

Data-driven intrinsic localized mode detection and classification in one-dimensional crystal lattice model

In this work we propose Support Vector Machine classification algorithms to classify one-dimensional crystal lattice waves from locally sampled data. Different learning datasets of particle displacements, momenta and energy density values are considered. Efficiency of the classification algorithms is further improved by two dimensionality reduction techniques: Principal Component Analysis and Locally Linear Embedding. Robustness of classifiers is investigated and demonstrated. Developed algorithms are successfully applied to detect localized intrinsic modes in three numerical simulations considering a case of two localized stationary breather solutions, a single stationary breather solution in noisy background and two mobile breather collision.

Dissipative localized states and breathers in phase-mismatched singly resonant optical parametric oscillators: Bifurcation structure and stability

We study the emergence of dissipative localized states in phase mismatched singly resonant optical parametric oscillators. These states arise in two different bistable configurations due to the locking of front waves connecting the two coexisting states. In one of these configurations, the bistability is mediated by the coexistence of two uniform states. Here the localized states are organized in a collapsed snaking bifurcation structure. Moreover, these states undergo oscillatory instabilities which lead to a breathing behavior. When the bistability is related to the coexistence of a uniform state and a spatially periodic pattern, localized states are organized in a bifurcation structure similar to the standard homoclinic snaking. Performing an exhaustive bifurcation analysis, we characterize in detail the previous structures, their linear stability, and the modification of their dynamics as a function of the control parameters of the system.

Changing the Dynamic Parameters of Localized Breather and Soliton Waves in the Sine-Gordon Model with Extended Impurity, External Force, and Decay in the Autoresonance Mode

Possibility of changing the dynamic parameters of localized breather and soliton waves for the sine-Gordon equation in the model with extended impurity, variable external force and dissipation was investigated using the autoresonance method. The model of ferromagnetic structure consisting of two wide identical layers separated by a thin layer with modified values of magnetic anisotropy parameter was taken as a basis. Frequency of external field is a linear function of time. The sine-Gordon equation (SGE) was solved numerically using the finite differences method with explicit scheme of integration. For certain values of the extended impurity parameters a magnetic inhomogeneity in the form of magnetic breather is formed when domain wall passes through it with constant velocity. The numerical simulation showed that using special variable force and small amplitude it is possible to resonantly increase the amplitude of breather. For each case of the impurity parameters values, there is a threshold value of the magnetic field amplitude leading to resonance. Geometric parameters of thin layer also have influence on the resonance effect — for decreasing layer width the breather amplitude grows more slowly. For large layer width the translation mode of breather oscillations is also excited. For certain parameters of extended impurity, a soliton can form. For a special type of variable field with frequency linearly dependent on time, soliton is switched to antisoliton and vice versa.

Existence of exponentially spatially localized breather solutions for lattices of nonlinearly coupled particles: Schauder’s fixed point theorem approach

The problem of showing the existence of localized modes in nonlinear lattices has attracted considerable efforts not only from the physical but also from the mathematical viewpoint where a rich variety of methods have been employed. In this paper, we prove that a fixed point theory approach based on the celebrated Schauder’s fixed point theorem may provide a general method to concisely establish not only the existence of localized structures but also a required rate of spatial localization. As a case study, we consider lattices of coupled particles with a nonlinear nearest neighbor interaction and prove the existence of exponentially spatially localized breathers exhibiting either even-parity or odd-parity symmetry under necessary non-resonant conditions accompanied with the proof of energy bounds of solutions.

Rogue breathers and rogue lumps on a background of dark line solitons for the Maccari system

The Maccari system is a nonlinear dynamical model for the description of two-dimensional rogue waves occuring in diverse physical settings. For this two-dimensional nonlinear system, the time localized breathers and doubly localized lumps termed as rogue breathers and rogue lumps on a background of dark line solitons are investigated. The rogue breather behaves as a breather appearing and disappearing on a background of two dark line solitons, and is generated from the resonant collision between a breather and two dark line solitons. The limit case of a rogue breather is a rogue lump, which is truly localized in time and in the two-dimensional space. The Nth-order rogue lumps on a background of (N+1) dark line solitons are constructed by employing the Kadomtsev-Petviashvili hierarchy reduction method combined with the Hirota bilinear method. These higher-order lumps arise from a background of dark line solitons and then disappear into the background again after living for a very short period of time. The rogue lumps possess the key features of two-dimensional rogue waves occurring in a plethora of physical settings.

Stability and instability of breathers in the U(1) Sasa–Satsuma and nonlinear Schrödinger models* *Partially funded by Product. CNPq Grant No. 305205/2016-1, IMUS and VI PPIT-US program reference I3C and CMM Conicyt PIA AFB170001 and Fondecyt 1150202.

We consider the Sasa–Satsuma (SS) and nonlinear Schrödinger (NLS) equations posed along the line, in 1 + 1 dimensions. Both equations are canonical integrable U(1) models, with solitons, multi-solitons and breather solutions Yang (2010 SIAM Mathematical Modeling and Computation). For these two equations, we recognize four distinct localized breather modes: the Sasa–Satsuma for SS, and for NLS the Satsuma–Yajima, Kuznetsov–Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior Grillakis et al (1987 J. Funct. Anal. 74 160–97). In this paper we find the natural H 2 variational characterization for each of them. This seems to be the first known variational characterization for these solutions; in particular, the first one obtained for the famous Peregrine breather. We also prove that Sasa–Satsuma breathers are H 2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang (2005 Chaos 15 037115). Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H 2 based Lyapunov functional, in the spirit of Alejo and Muñoz (2013 Commun. Math. Phys. 324 233–62), extended this time to the vector-valued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma–Yajima, Peregrine and Kuznetsov–Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in Muñoz (2017 Proyecciones (Antofagasta) 36 653–83).

Discrete magnetic breathers in monoaxial chiral helimagnet

We analyze spatially localized breather excitations for the model of a discrete Heisenberg spin chain, which includes the antisymmetric exchange interaction and single-ion anisotropy of the easy-plane type. In a finite size chain the breather modes may be indexed by number of embedded kink-antikink pairs forming the regular breather lattice. The influence of the antisymmetric exchange on properties of the discrete breather modes is examined.

Breather solutions of the cubic Klein–Gordon equation

We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation which are weakly localized in space. Various families of such ‘breather’ solutions are shown to bifurcate from any given nontrivial stationary solution. The construction of weakly localized breathers in three space dimensions is, to the author’s knowledge, a new concept and based on the reformulation of the cubic Klein–Gordon equation as a system of coupled nonlinear Helmholtz equations involving suitable conditions on the far field behavior.

Mobile Dissipative Breathers in a Chain of Nonlinear Oscillators

In a numerical experiment, it is found that at least three localized stationary breather type modes can exist in a chain of Rayleigh oscillators with a nonlinear coupling between them: immobile, “slow,” and “fast” dissipative breathers. The dynamics of synchronous motions of chain elements depends on the relation of characteristic time scales of the system related to the frequency of the oscillators and rigidity of the coupling between them; it is multistable.

Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry

We present experimental evidence of formation and persistence of localized waves, breathers, and solitons, occurring in a random sea state and uniformly traveling over non-uniform bathymetry. Recent studies suggest connections between breather dynamics and irregular sea states and between extreme wave formation and breathers, random sea states, or non-uniform bathymetry individually. In this paper, we investigate the joint connection between these phenomena, and we found that breathers and deep-water solitons can persist in more complex environments. Three different sets of significant heights have been generated within a Joint North Sea Wave Observation Project wave spectrum, and the wave heights were recorded with gauges in a wave tank. Statistical analysis was applied to the experimental data, including the space and time distribution of kurtosis, skewness, Benjamin–Feir index, moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random wave field and traveling over shoals, valleys, and slopes were compared with exact solutions of the nonlinear Schrödinger equation with a good match, demonstrating that these localized waves have the same structure as deep-water breathers. We identify the formation of rogue waves at moments and over regions where the kurtosis and skewness have local maxima. These results provide insights for understanding of the robustness of Peregrine and higher-order Akhmediev breathers, Kuznetsov–Ma solitons, and rogue waves, and their occurrence in realistic oceanic conditions, and may motivate analogous studies in other fields of physics to identify limitations of exact weakly nonlinear models in non-homogeneous media.

Parametric localized patterns and breathers in dispersive quadratic cavities

We study the formation of localized patterns arising in doubly resonant dispersive optical parametric oscillators. They form through the locking of fronts connecting a continuous-wave and a Turing pattern state. This type of localized pattern can be seen as a slug of the pattern embedded in a homogeneous surrounding. They are organized in terms of a homoclinic snaking bifurcation structure, which is preserved under the modification of the control parameter of the system. We show that, in the presence of phase mismatch, localized patterns can undergo oscillatory instabilities which make them breathe in a complex manner.

Spherically localized discrete breathers in bcc metals V and Nb

Discrete breathers (DBs), also called intrinsic localized modes (ILMs), are spatially localized, large-amplitude vibrational modes in nonlinear lattices. Rod-like discrete breathers (DBs) have been reported in fcc Ni and bcc Nb by Haas et al. in 2011 based on molecular dynamics simulations. Here we use a general approach to find new type DBs in bcc V and Nb. In this approach, we firstly find the lattice symmetry dictated exact solutions to the equations of atomic motion in the form of delocalized nonlinear vibrational modes (DNVMs). Secondly, a localizing function with spherical symmetry is imposed over the DNVMs. Parameter of the localizing function is chosen such that the obtained DB has a long lifetime. The results presented in this work demonstrate that pure metals can support a variety of DBs. Interatomic potentials have a strong effect on the DB lifetime. Maximal DB lifetime in V is two orders of magnitude larger than that in Nb. The results of this study are interesting for the theory of DBs, and also they will help to better understand the impact of DBs on the physical properties of metals.