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Korteweg-de Vries-type equations have occurred in the fields of planetary oceans, atmospheres, cosmic plasmas and so on, while nonlinear evolution equations with the variable coefficients have provided a realistic perspective on the inhomogeneities of media and non-uniformities of boundaries. In this paper, we investigate a (3+1)-dimensional Korteweg-de Vries equation with the time-dependent coefficients in a fluid. Based on the Hirota method, we obtain a bilinear form via the binary Bell polynomial approach. Based on the bilinear form, we derive the N-soliton, breather and periodic-wave solutions, where N is a positive integer. Besides, we investigate the asymptotic behaviors of the breather and periodic-wave solutions. Breather waves and periodic waves are graphically displayed. Finally, relation between the periodic-wave solutions and one-soliton solutions is discussed. This paper provides an intuitive understanding for the nonlinear phenomena of those obtained solutions, and those nonlinear phenomena have potential application value in fluid dynamics and other fields. [ABSTRACT FROM AUTHOR]

This article describes the bilinear form, biliear Bäcklund transformation and Lax pair of the geophysical Boussinesq equation using the Bell polynomial approach. The integrability of the said equation is asserted in the sense of the Lax pair. The Darboux transformation is employed to produce periodic solutions as well as single and N-complexiton-type solutions. Several steps of the Hirota bilinear technique are used to illustrate the flow characteristics of multi-solitons, higher-order breathers and rogue waves. Every sort of wave dynamics response to the Coriolis force is carefully studied. [ABSTRACT FROM AUTHOR]

This study explores various wave phenomena related to the concatenation model, which is characterized by the inclusion of the Kerr law of nonlinearity in birefringent fibers. Several distinct auxiliary functions and logarithmic transformation are utilized to formulate various analytical solutions, including multi-wave solutions, two solitary wave solutions, breather waves, periodic cross kink solutions, Peregrine-like rational solutions, and three-wave solutions. To demonstrate the influence of different parameters on the interaction of the obtained solutions, some figures are provided to vividly display these transmission and interaction characteristics. [ABSTRACT FROM AUTHOR]

In the presence of spatio-temporal dispersion, perturbation terms of the Hamiltonian type as well as multiplicative white noise, analytical investigation of the concatenation model having the Kerr law of nonlinearity is carried out in this work. The Cole–Hopf transformation and direct assumptions with arbitrary functions are utilized to determine several analytic solutions to the governing equation, including multi-wave, two solitary wave, breather, periodic cross kink, Peregrine-like rational, and three-wave solutions. The parameter constraints that serve as the requisite condition for the existence of these wave solutions are also identified. In order to explore and illustrate the propagation and dynamical behaviors of some solutions reported in this research, 3D graphics and their corresponding contour plots are included. Results of this paper may be useful for the experimental realization of certain nonlinear waves in optical fibers and further understanding of their propagation dynamics. [ABSTRACT FROM AUTHOR]

Fluid mechanics concerns the mechanisms of liquids, gases and plasmas and the forces on them. We aim to investigate a (2 + 1) -dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics in this paper. With the help of the Painlevé analysis, we find that the above system has Painlevé-integrable property. A set of the auto-Bäcklund transformations and some solutions are derived by the virtue of the truncated Painlevé method. We obtain certain bilinear forms via some seed solutions. According to the mentioned bilinear form, we derive the multiple-soliton solutions on some nonzero backgrounds. Based on the soliton solutions and conjugation transformations, the higher-order breather solutions on certain nonzero backgrounds have been obtained. Via some conjugation transformations, hybrid solutions formed from the breathers and solitons on certain nonzero backgrounds have been derived. We also graphically show the interactions between those solitons and breathers. [ABSTRACT FROM AUTHOR]

INTRODUCTION. Acquired pneumonia is a severe medical condition that is addressed as life-threating issue requiring intensive care. The Medical Breather device can activate and strengthen both the inspiratory and expiratory muscles, so it can be useful for patients with pneumonia. AIM. To investigate the breather effect on length of hospital stay in patients with pneumonia. MATERIALS AND METHODS. Sixty participants diagnosed with acquired pneumonia “30 women, 30 men stayed in hospital in ICU for two weeks; aged 30–40 years old” selected from chest department of Kasr Al-Aini Intensive Care Unit (ICU) at Cairo University. They were randomly allocated into equal groups; Group A received respiratory training via incentive spirometer, and traditional chest physiotherapy; and Group B received respiratory training via Breather, and traditional chest physiotherapy, both received 3 session daily/2 weeks. Diaphragmatic excursion, Respiratory Distress Observation Scale, and ICU discharge were assessed before and after the treatment. RESULTS. Both groups revealed significant improvement after the treatment, while Breather group showed a high significant increase in pH 1.23 %, PaO2 11.79 %, SaO2 6.1 %, and diaphragmatic excursion by 36.97 %, also decrease in PaCO2 2.78 %, RDOS 39.06 % and NEWS2 by 50.72 % in comparison to incentive spirometer group that recorded significant increase in pH 0.68 %, PaO2 6.69 %, SaO2 by 2.66 %, and diaphragmatic excursions by 8.15 %, also significant decrease in PaCO2 12.12 %, RDOS 15.01 % and NEWS2 by 20.93 %. HCO3 revealed no significant difference post treatment (p 0.05). DISCUSSION. Breather usage in inspiratory musculatures training (IMT) gained Maximum Inspiratory Pressure (Pimax) significant improvement. IMT enforces both diaphragm and accessory respiratory musculatures. Probably functional capabilities improvements based on enhanced respiratory musculatures’ both endurance and strength that improve pulmonary oxygen uptake thus minimize dyspnea severity. Respiratory muscles training program improves not only cognitive function. Moreover, IMT could be addressed as a prime component of respiratory training in combine with expiratory one that is why whom has preserved pulmonary function. CONCLUSION. Breather as a respiratory training technique has remarkable results in reducing hospital stays in patients with acquired pneumonia, and significant positive effects on diaphragmatic function, oxygenation levels. Therefore, it is recommended to use Breather for routine acquired pneumonia care. REGISTRATION: Clinicaltrials.gov identifier: No NCT06062862; registered April 30, 2022.

Interactions of nonlinear waves play a significant role in physical systems. In this work, we attain the interactions of breathers and rogue wave for the coupled Lakshmanan–Porsezian–Daniel equation. Through the Darboux transformation, we succeed the interactional solutions consisting of different types of breathers and rogue wave. In particular, it can be observed that distinct parameters a 1 , a 2 , δ and λ 1 make the interaction properties, structures and energy conversion of breathers and dramatically change. Moreover, we exhibit the striking dynamics of interactional solutions based on three-dimensional figures. These results are helpful for the study of nonlinear waves in coupled integrable systems. [ABSTRACT FROM AUTHOR]

The study delves into the behavior of cos-Gaussian breathers navigating through a highly nonlocal medium, with a spotlight on their width and intensity variations under different parameter adjustments. The mathematical formulas describing the characteristics of cos-Gaussian beams have been derived. By controlling the parameters, the transverse width of cos-Gaussian beams can exhibit various variations, reflecting various mode transformations. We scrutinized how alterations in cosine parameters, offset distance, and initial energy manifest in the breather's width and shape during propagation. Our observations underscore the significant influence of these parameters on the dynamic evolution and morphology of cos-Gaussian breathers, thereby unveiling the potential application of using parameters to control the dynamic characteristics of cos-Gaussian beam transmission. [ABSTRACT FROM AUTHOR]

We are interested in stability results for breather solutions of the 5th, 7th and 9th order mKdV equations. We show that these higher order mKdV breathers are stable in H²(R), in the same way as classical mKdV breathers. We also show that breather solutions of the 5th, 7th and 9th order mKdV equations satisfy the same stationary fourth order nonlinear elliptic equation as the mKdV breather, independently of the order, 5th, 7th or 9th, considered. [ABSTRACT FROM AUTHOR]

Soliton dynamics often exhibit a rich diversity and complexity, emerging from myriad combinations of nonlinearities and dispersions within nonlinear dynamical systems. This endeavor contributes to the novel exploration of intricate soliton interactions. In this paper, we investigate a multi-component Maccari system which can be used to depict optical pulse motions in multi-mode optical fibers. By employing the bilinear method, we obtain the system's analytical second-order solutions involving abundant parameters. Under taking suitable parameter settings, we observe some novel soliton interaction dynamics: soliton resonances from local to global ranges, transitions from soliton molecules to breathers, semi-elastic soliton collisions and soliton bifurcations (namely solitons' fission and fusion). Especially, all solitons in resonances, molecules and bifurcations hold stable propagation states. The findings also reveal new energy transition mechanism during soliton interactions by semi-elastic collisions and soliton bifurcations. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the (2+1)- dimensional coupled nonlinear Schrödinger equations with variable coefficients which are used to describe the propagation of beams in inhomogeneous and nonlinear birefringent fibers, taking into account the components in the two polarized directions. To tackle this problem, we employ the Hirota bilinear method. Multiple solitons, rogue waves, breather waves and their interaction solutions relating to the suitable choice of time-dependent coefficients are obtained. Through manipulating the relevant parameters, the propagation control and evolution are investigated for them. Several interesting transition phenomena are revealed, such as, the transitions from the bright soliton to breather, from bright rogue to periodic humps of rogue wave structures overlay on the one- and two-peak bright soliton and from the plane wave background to the periodic ones. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the variable coefficient nonlocal Fokas–Lenells equation. On the basis of the Lax pair, the infinitely-many conservation laws and Nth-fold Darboux transformation are constructed. Depending on zero seed solution, soliton solutions are derived via the Darboux transformation. Based on nonzero seed solution, breather solutions and rogue wave solutions are obtained. The behaviors of solutions are clearly analyzed graphically. The influences of variable coefficient for solutions are discussed. The different profiles of solitons, breathers and rogue waves are observed via selecting different variable coefficients. Furthermore, the interaction of solitons and the interaction of breathers for the variable coefficient nonlocal Fokas–Lenells equation are both elastic. [ABSTRACT FROM AUTHOR]

Under investigation is a concatenation model having Kerr law of nonlinearity. This model encompasses nonlinear Schrödinger's equation, Lakshmanan–Porsezian–Daniel model and Sasa–Satsuma equation. A variety of analytical solutions are recovered with the help of ansatz functions and logarithmic transformation. These solutions that emerge from ansatz functions include multi-waves, two solitary wave solutions, breather waves, periodic cross kink solutions as well as Peregrine-like rational solutions. Propagation characteristics of the solutions revealed are also displayed with numerical simulations. [ABSTRACT FROM AUTHOR]

In this work, under investigation is a defocusing Lakshmanan–Porsezian–Daniel system governed by a four-order nonlinear Schrödinger-like equation arising from optical fibers. Via applying the Darboux transformation technique, we first construct new analytical solutions for the system. Then, by setting and adjusting the spectrum parameters involved in the solutions, some interesting and novel optical soliton structures and propagation properties are found out, such as, weakly local and strongly local soliton resonances, global soliton resonances, soliton molecules, breathers with peak-like and loop-like patterns. The study also reveals the formation mechanisms from local soliton resonances to breathers, from soliton molecules to breathers. All these waves possess stable propagation profiles, e.g, shapes, speeds and amplitudes. [ABSTRACT FROM AUTHOR]

We have obtained the first-order solution of a three-coupled nonlinear Schrödinger equation based on the modified Darboux transformation. In addition, we have derived an expression for the distance between the rogue wave and breather. We have find that the amplitude of the rogue wave, the period of the breather, the distance between the rogue wave and breather, the transformation between the bright and dark rogue wave, and the transformation between the soliton and breather which are all affected by the values of the free parameters. [ABSTRACT FROM AUTHOR]

In this paper, based on the modified Darboux transformation, a new first-order solution of coupled fourth-order nonlinear Schrödinger equation (cNLS) is constructed. The amplitude of rogue wave, distance of the breather and the rogue wave can be changed if we adjust parameter d 1 . With the adjustment of the parameter c 2 , the breather and the rogue wave can be converted into each other, and the direction of propagation of the breather can be changed. When the initial wave height takes different values, images of the breather and the rogue wave as well as soliton-like and rogue waves can be presented. [ABSTRACT FROM AUTHOR]

In this article, we investigate the dynamical interaction behavior of a short pulse equation in optical fibers with fast-varying packets. We systematically unearth the interaction dynamics between solitons, breathers, and their hybrid forms. Using the bilinear method, we explicitly calculate the first- to fourth-order solutions. We categorize the solutions into three classes based on their dispersion coefficients: stripe-loop-like soliton, breather, and their hybrid form. We observe the existence of bright and dark solitons. Additionally, a breather may consist of periodical peak-trough waves and periodical kink-loop-like waves. As the order of the solutions increases, there are abundant and complicated interaction behaviors for the solitons, breathers, and their hybrid forms due to these rich patterns. [ABSTRACT FROM AUTHOR]

Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of the kinks in the sine-Gordon model with “impurities” (or spatial inhomogeneity of the periodic potential). Methods. Using the method of collective variables for the case of three identical point impurities located at the same distance from each other, a system of differential equations is obtained. Resulting system of equations makes it possible to describe the dynamics of the kink taking into account the excitation of localized waves on impurities. To analyze the dynamics of the kink in the case of extended impurities, a numerical finite difference method with an explicit integration scheme was applied. Frequency analysis of kink oscillations and localized waves calculated numerically was performed using a discrete Fourier transform. Results. For the kink dynamics, taking into account the excitation of oscillations in modes, a system of equations for the coordinate of the kink center and the amplitudes of waves localized on impurities is obtained and investigated. Significant differences are observed in the dynamics of the kink when interacting with a repulsive and attractive impurity. The dynamics of the kink in a model with three identical extended impurities, taking into account possible resonant effects, was solved numerically. It is established that the found scenarios of kink dynamics for an extended rectangular impurity are qualitatively similar to the scenarios obtained for a point impurity described using a delta function. All possible scenarios of kink dynamics were determined and described taking into account resonant effects. Conclusion. The analysis of the influence of system parameters and initial conditions on possible scenarios of kink dynamics is carried out. Critical and resonant kink velocities are found as functions of the impurity parameters.

Under investigation in this paper is the integrable Lakshmanan–Porsezian–Daniel (LPD) equation, which was proposed as a model for the nonlinear spin excitations in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin. Our main purpose was to construct soliton and breather solutions on the nonconstant background for the integrable local and nonlocal LPD equations. Firstly, the Bäcklund transformations are constructed based on the pseudopotential of equations. Secondly, starting from the nonconstant initial solution sech and applying the obtained transformation, various nonlinear wave solutions of the local LPD equation are provided, including the time-periodic breather, W-shaped soliton, M-type soliton and two-soliton solutions, the elastic interactions between the two-soliton solutions are shown and the relationship between parameters and wave structures is discussed. Thirdly, beginning with the nonconstant initial solutions sech and tanh , the time-periodic breather, bell-shaped one-soliton and anti-bell-shaped one-soliton solutions of the nonlocal LPD equation are generated and these solutions possess no singularity. What is more, the time-periodic breather solutions exhibit the x-periodic background and double-periodic background, which is different from the previous results. The corresponding dynamics of these solutions related to the integrable local and nonlocal LPD equations are illustrated graphically. The results in this paper might be helpful for us to understand the nonlinear characteristics of magnetic materials. [ABSTRACT FROM AUTHOR]

In this paper, we construct an ancient solution by using a given shrinking breather and prove a no shrinking breather theorem for noncompact harmonic Ricci flow under the condition that Sic : = Ric − α ∇ ϕ ⊗ ∇ ϕ is bounded from below. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation which provides overwhelming support for studying the dynamics of high-dimensional nonlinear wave equations. The bilinear form of the equation is obtained based on the Hirota bilinear method, and the N-soliton solutions composed of the higher-order breather, periodic line wave and the mixed forms are constructed. Then, the rational and semi-rational solutions of the equation were acquired by using complex conjugate parameter relations and the long-wave limit method, which mainly consisted of high-order solitons, lumps, breathers and their mixed forms. We analyze the effect of the coefficients of space and time variables on the interaction of solutions to bilinear equations. By classifying these coefficients, we find that these coefficients change the interaction of the solutions by affecting the velocity, position, and trajectory of the waves. In order to describe the dynamics of solutions with different parameters more directly, the time evolution plots and density plots are presented, and the appearance and movement characteristics of the solutions are analyzed. [ABSTRACT FROM AUTHOR]

The Kuralay equation, as a Heisenberg ferromagnet-type equation, is of great significance for describing nonlinear phenomena in magnets. For the Kuralay-IIA equation, based on the constructed N -fold Darboux transformation, we derive its one-fold and two-fold rogue wave solutions, one-breathers and two-breathers including the Akhmediev two-breathers. By adding shifts on the time variable, we further separate the Akhmediev two-breathers into two one-breathers. In addition, the semi-rational solutions consisting of the rogue wave and one-breather solutions are also derived. The diagrams of the obtained solutions are plotted to analyze their dynamic features. [ABSTRACT FROM AUTHOR]

In this paper, the spatial discrete Hirota equation is investigated by Darboux–Bäcklund transformation. Firstly, the pseudopotential of the spatial discrete Hirota equation is proposed for the first time, from which a Darboux–Bäcklund transformation is constructed. Comparing it with the corresponding onefold Darboux transformation, we find that they are equivalent because there is no difference except for a constant times. We believe that this equivalence may hold universal if these two transformations are all derived from the same discrete spectral problem and using the similar technique in the references. Secondly, starting from vanishing and plane wave backgrounds, a variety of nonlinear wave solutions, including bell-shaped one-soliton, three types of breathers, W-shaped soliton, periodic solution and rogue wave are given, and the relevant dynamical properties and evolutions are illustrated by plotting figures. The relationship between parameters and solutions' structures is studied in detail, and the related method and technique can also be extended to other nonlinear integrable equations. Finally, we show that the continuum limit of breather and rogue wave solutions of the spatial discrete Hirota equation yields the counterparts of the Hirota equation. The results in this paper might be useful for understanding some physical phenomena in nonlinear optics. [ABSTRACT FROM AUTHOR]

Investigated in this paper is an extended (3+1)-dimensional Kadomtsev-Petviashvili equation. We determine the N-soliton solutions of that equation via an existing bilinear form, and then construct the Mth-order breather and Hth-order lump solutions from the N-soliton solutions using the complex conjugated transformations and long-wave limit method, where N, M, and H are the positive integers. In addition, we develop the hybrid solutions composed of the first-order breather and one soliton, the first-order lump and one soliton, as well as the first-order lump and first-order breather. Through those solutions, we demonstrate the (1) one breather or lump, (2) interaction between the two breathers or lumps, (3) interaction between the one breather and one soliton, (4) interaction between the one lump and one soliton, and (5) interaction between the one lump and one breather. We observe that the amplitude, shape, and velocity of the one breather or lump remain unchanged during the propagation. We also find that the amplitudes, shapes, and velocities of the solitons, breathers, and lumps remain unchanged after the interactions, suggesting that those interactions are elastic. [ABSTRACT FROM AUTHOR]

We consider the modified Korteweg–de Vries equation, and prove that given any sum P of solitons and breathers (with distinct velocities), there exists a solution p such that p(t) - P(t) → 0 when t → + ∞, which we call multi-breather. In order to do this, we work at the H² level (even if usually solitons are considered at the H¹level). We will show that this convergence takes place in any Hs space and that this convergence is exponentially fast in time. We also show that the constructed multi-breather is unique in two cases: in the class of solutions which converge to the profile P faster than the inverse of a polynomial of a large enough degree in time (we will call this a super polynomial convergence), or when all the velocities are positive (without any hypothesis on the convergence rate) [ABSTRACT FROM AUTHOR]

Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of coupled localized nonlinear waves in the sine-Gordon model with impurities (or spatial inhomogeneity of the periodic potential). Methods. Using the analytical method of collective coordinates for the case of the arbitrary number the same point impurities on the same distance each other, differential equation system was got for localized waves amplitudes as the functions on time. We used the finite difference method with explicit scheme for the numerical solution of the modified sine-Gordon equation. We used a discrete Fourier transform to perform a frequency analysis of the oscillations of localized waves calculate numerically. Results. We found of the differential equation system for three harmonic oscillators with the elastic connection for describe related oscillations of nonlinear waves localized on the three same impurity. The solutions obtained from this system of equations for the frequencies of related oscillation well approximate the results of direct numerical modeling of a nonlinear system. Conclusion. In the article shows that the related oscillation of nonlinear waves localized on three identical impurities located at the same distance from each other represent the sum of three harmonic oscillations: in-phase, in-phase-antiphase and antiphase type. The analysis of the influence of system parameters and initial conditions on the frequency and type of associated oscillations is carried out.

In this paper, we present a direct approach to various solutions of the modified Korteweg‐de Vries‐sine Gordon equation, which is a versatile model emerging in many physics and mathematics contexts. The direct approach is based on Cauchy matrix and Sylvester equation. The obtained solutions can be classified with respect to eigenvalues of certain matrix. We present eigenvalue structures that give rise to kinks, breathers, soliton molecules, multiple‐pole solutions, and so forth. Some obtained solutions are illustrated. [ABSTRACT FROM AUTHOR]

Preventive maintenance in the transformer is performed through a differential relay protection system, and it protects the transformer from internal and external faults. However, the Current Transformer (CT) in the differential protection system mal-operates during inrush currents. CT saturates due to magnetizing inrush currents and causes false tripping of the differential relays. Moreover, identification of tripping in protection relay either due to inrush current or internal faults needs to be diagnosed. For the above problem, continuous monitoring of transformer breather and CT terminals with thermal camera helps detect the tripping in relay due to inrush or internal fault. The transformer's internal fault leads to high breathing process in the transformer breather, never for inrush currents. During inrush currents, CT temperature is increased. Continuous monitoring of breather and CT of the transformer through thermal imaging and radiometric pixels detect the causes of CT saturation and differentiates maloperation. Hybrid wavelet threshold image analytics (HWT-IA) based radiometric pixels analysis of the transformer breather and CT after de-noising provides an accurate result of about 95% for identification of the false tripping of differential protection system of transformer. [ABSTRACT FROM AUTHOR]

The first- and second-order localized waves for a variable-coefficient coupled Hirota equation describe the vector optical pulses in inhomogeneous optical fiber and are investigated via generalized Darboux transformation in this work. Based on the equation's Lax pair and seed solutions, the localized wave solutions are calculated, and the dynamics of the obtained localized waves are shown and analyzed through numerical simulation. A series of novel dynamical evolution plots illustrating the interaction between the rogue waves and dark-bright solitons or breathers are provided. It is found that functions have an influence on the propagation of shape, period, and velocity of the localized waves. The presented results contribute to enriching the dynamics of localized waves in inhomogeneous optical fiber. [ABSTRACT FROM AUTHOR]

In this paper, a coupled mixed derivative nonlinear Schrödinger system, which describes the short pulses in the femtosecond or picosecond regime of a birefringent optical fiber, is investigated. Based on the known Nth-order breather solutions, we derive the first-order breathers and investigate their properties, e.g., velocities and peak amplitudes, where N is a positive integer. Then, two kinds of the second-order breathers are presented. We construct the Nth-order semirational solutions, with only one spectral parameter involved. Based on the obtained Nth-order semirational solutions, we analytically investigate and graphically illustrate the vector degenerate breathers, rogue and breather-rogue waves. We discuss how certain parameters, e.g., the nonlinear coefficients, affect the shapes of the degenerate breathers, rogue and breather-rogue waves. [ABSTRACT FROM AUTHOR]

In this work, we deal with the initial value problem of the 5th-order Gardner equation in R , presenting the local well-posedness result in H 2 (R) . As a consequence of the local result, in addition to H 2 -energy conservation law, we are able to prove the global well-posedness result in H 2 (R) . Finally as a direct application, we prove that some globally defined functions, e.g. breather solutions of 5th order Gardner equation, are H 2 (R) stable. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to learn N-soliton, M-lump, breather solutions and interaction solutions for a KdV equation with variable coeffificients. Using the Hirota bilinear method, one-soliton, two-soliton, three-soliton and N-soliton are obtained. By taking the conjugate parameters for two-soliton amd four-soliton, one-breather and two-breather waves are obtained, respectively. The long wave limit technique is applied to two-soliton, four-soliton and six-soliton, one-lump, two-lump and three-lump solutions are obtained, respectively. Furthermore, the interaction solutions are constructed, including one-breather and one-soliton, one-breather and two-soliton, one-lump and one-soliton, one-lump and two-soliton. In order to discussing the dynamical properties of the above solutions, some 3D-plots, 2D-plots, contour plots and density plots of these solutions are given.

The higher-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions of the Hirota–Maccari (HM) system by virtue of the Kadomtsev–Petviashvili (KP) hierarchy reduction method are investigated in this work. Through analyzing the structural characteristics of periodic-wave solutions, we attain the quasi-periodic W(M)-shaped waves and two kinds of breathers. The mixed solutions that consist of the quasi-periodic W(M)-shaped waves and breathers are constructed. Further, by taking the long wave limit on the periodic-wave solutions, the semi-rational solutions are derived, which illustrate the interaction of the rational soliton, lump, quasi-periodic wave and breather. Characteristics of these mixed solutions are discussed graphically and the corresponding generating conditions are given. Especially, a new bound-state interaction composed of lump and breather is generated under the velocity resonance mechanism. This newfangled pattern is a beautiful phenomenon for the HM system. [ABSTRACT FROM AUTHOR]

In this paper, a new extended (3 + 1)‐dimensional generalized Kadomtsev–Petviashvili (KP)–Boussinesq equation, with two additional terms utt$$ {u}_{tt} $$ and uxz$$ {u}_{xz} $$, is proposed and investigated. This new equation models both left and right going waves like the Boussinesq equation and describes the transmission of tsunami waves at the bottom of the ocean and nonlinear ion‐acoustic waves in the magnetized dusty plasm. We have constructed one set of breathers and first order periodic waves from the two‐soliton solution. Moreover, the four‐soliton solution consists of two sets of breathers, second order periodic waves, and a hybrid of breathers and periodic line waves. Then, we introduce two kinds of "long wave" limits to obtain the rational and semi‐rational solutions for N=2,3,4$$ N=2,3,4 $$. Under specific parametric constraints, the obtained rational and semi‐rational solutions are nonsingular. The rational solution can be classified as first order line rogue wave, single breather, second order line rogue wave, double breather, a hybrid of breather and line rogue wave, a hybrid of breather, and single soliton. The semi‐rational solution can be classified as the first and second order kink‐shaped rogue wave, a hybrid of breather and one (two) soliton(s), a hybrid of a set of breathers, and a single soliton (breather). In addition, we give Theorem 2.1 for the higher order rational solutions. [ABSTRACT FROM AUTHOR]

By the means of reconstructing the Darboux transformation to the Manakov system, we obtain the new explicit solutions composed of a rogue wave and a breather for the system. Via controlling the parameter involved in the solutions, we reveal a novel 'firewall' effect during the rogue wave and breather interactions. There are three cases of inelastic collisions and one type of semi-elastic collision between the rogue wave and breather. Particularly, either of the rogue wave or breather cannot cross over another during the collision. [ABSTRACT FROM AUTHOR]

In the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two‐component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single‐component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one‐ and two‐soliton for bright, dark soliton and breather solutions are analyzed in details. [ABSTRACT FROM AUTHOR]

The ( 2 + 1 )-D complex modified Korteweg–de Vries (cmKdV) equations are investigated with the aid of the Darboux transformation method. Through the limits λ 2 k - 1 → λ 0 = - a 2 + c i (k = 1 , ... , m , m ⩽ n - 1) , the order-n semi-rational solutions are obtained. The order-2 semi-rational solutions and order-3 semi-rational solutions are analyzed in detail. By changing different parameters l j , different semi-rational solutions are deduced, including rogue wave interaction with the periodic wave or breather and lump interaction with the periodic wave or breather. The dynamical properties of these solutions are discussed, which indicates that these interactions are elastic collisions. In terms of application, these semi-rational solutions will be valuable in modeling physical problems. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear wave solutions for Gross–Pitaevskii equation on the periodic wave background are investigated by Darboux-Bäcklund transformation, from which the soliton and breather wave solutions on the Jacobi elliptic cn and dn functions backgrounds are derived. The corresponding evolutions and dynamical properties of nonlinear wave solutions under different parameters are discussed. These results reported in this paper may raise the possibility of related experiments and potential applications in nonlinear science fields, such as nonlinear optics, oceanography and so on. [ABSTRACT FROM AUTHOR]

Hopf‐type bifurcation dynamics, a universal phenomenon existing in numerous physical systems, has recently been observed in mode‐locked erbium‐doped fiber lasers with anomalous or normal net dispersion. This study demonstrates the real‐time experimental observation of double‐Hopf‐type breathers in an all‐normal dispersion all‐polarization maintaining ytterbium‐doped fiber laser instead. It is shown that the breather frequency can be modulated periodically by the additional oscillation with increasing amplitude in response to increased pump power until a stationary dissipative soliton is formed. The possible explanation of the observed double‐Hopf‐like bifurcation dynamics is discussed by exploring the numerical approach that combines the interplay of the population inversion in the laser medium with the pulses energy. The results provide additional building blocks for further understanding laser physics and can help in optimizing fiber cavity designs. [ABSTRACT FROM AUTHOR]

In this paper, the extended nonlinear Schrödinger equation with higher-order operators, which can be diffusely used to describe the pulses propagating along an optical fibre, is under investigation. By means of the generalized Darboux transformation, we present the interactional solutions composed of the breather and rogue wave. Furthermore, regulating the coefficients of higher-order operators results in miscellaneous patterns of the interactions between the breather and rogue wave. Especially, the solutions in the extended nonlinear Schrödinger equation are more abundant than ones in the classical nonlinear Schrödinger equation. The kinetics of interactional solutions are elucidated graphically. The method provided in this paper can also be adopted to construct interactional solutions of other higher-order nonlinear integrable equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the resonant collisions among different types of localized solitary waves in the (2+1)-dimensional Hirota–Satsuma–Ito equation, which are described by N-soliton solutions constructed using bilinear method. Through the asymptotic analysis and limit treatment of the phase shift of these localized waves, the elastic collisions among different localized waves can be transformed into resonant collisions. Hereby, we study the resonant collision between a breather/ lump and a bright line soliton and find two collision situations: (i) the breather is semi-localized in space and the shape of the breather is not localized during the propagation and (ii) the lump wave generates from the bright line wave. At the same time, we investigate the resonant collision between a breather/lump and two bright line solitons. In these evolution processes, we also gain two dynamical behaviors: (iii) the breather is always localized in space and the shape of the breather is not localized during the propagation, and (iv) the lump wave appears from a bright line soliton and then disappears into the other bright line soliton. Localized wave and interaction solutions of the nonlinear wave models have a great impact on oceanography and physics. The results may be useful in researching the physical phenomena in shallow water waves and nonlinear optics. [ABSTRACT FROM AUTHOR]

In this paper, a generalized long-wave short-wave resonance interaction system, which describes the nonlinear interaction between a short-wave and a long-wave in fluid dynamics, plasma physics and nonlinear optics, is considered. Using the Hirota bilinear method, the general N-bright and N-dark soliton solutions are deduced and their Gram determinant forms are obtained. A special feature of the fundamental bright soliton solution is that, in general, it behaves like the Korteweg-deVries soliton. However, under a special condition, it also behaves akin to the nonlinear Schrödinger soliton when it loses the amplitude-dependent velocity property. The fundamental dark-soliton solution admits anti-dark, gray, and completely black soliton profiles, in the short-wave component, depending on the choice of wave parameters. On the other hand, a bright soliton-like profile always occurs in the long-wave component. The asymptotic analysis shows that both the bright and dark solitons undergo an elastic collision with a finite phase shift. In addition to these, by tuning the phase shift regime, we point out the existence of resonance interactions among the bright solitons. Furthermore, under a special velocity resonance condition, we bring out the various types of bright and dark soliton bound states. Also, by fixing the phase factor and the system parameter β , corresponding to the interaction between long and short wave components, the different types of profiles associated with the obtained breather solution are demonstrated. [ABSTRACT FROM AUTHOR]

The KP hierarchy reduction method is one of the most reliable and efficient techniques for determining exact solitary wave solutions to nonlinear partial differential equations. In this paper, according to the KP hierarchy reduction technique, rational and some other semi-rational solutions to the (2 + 1)-dimensional Maccari system are investigated. It is shown that two different types of breathers can be derived, and under appropriate parameter constraints, they can be reduced to some well known solutions, involving the homoclinic orbits, dark soliton or anti-dark soliton solution. For the dark and anti-dark solution, its interaction is similar to a resonance soliton. Furthermore, by using a limiting technique, we derive two kinds of rational solutions, one is the lump and the other one is the rogue wave. After constructing these solutions, we further discuss the interactions between the obtained solutions. It is interesting that we obtain a parallel breather and a intersectional breather, which seems very surprising. Finally, we also provide a new three-state interaction, which is composed by the dark-soliton, rogue wave and breather and has never been provided for the Maccari system. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to investigate the various of interaction solutions to the (2+1)-dimensional variable-coefficient Sawada–Kotera equation. Analytical one-, two-, three- and four-soliton solutions of this equation are constructed based on its Hirota bilinear form. Through granting appropriate parameters and coefficients, special cases lead to resonance X-type soliton, S-type, U-type or periodic-type soliton solutions. The interaction solutions of multiple line solitons, between S-type line soliton and breather, U-type or S-type line soliton and lump wave, two U-type or S-type or periodic-type breathers, two S-type lump waves, among U-type or S-type two line solitons and one lump are obtained analytically, and some figures are provided with a better understanding of the dynamic behavior. We are confident that the results obtained in this paper are novel, which may be helpful to study other higher-dimensional nonlinear evolution equations

In this paper, the nonlinear propagation dynamics of circular–linear edge dislocation beams (CLED beams) in nonlocal media are studied theoretically. The transport formulae of CLED beams in nonlocal nonlinear media is derived. The evolution of the beam width, curvature, and intensity distribution are discussed in detail. Two different incident cases of CLED beams, namely beam waist and non-beam waist incidences, are discussed and compared. It is found that the off-axial distance can effectively adjust the position of linear edge dislocation and play a key role on beam propagation. The initial beam power and non-beam waist distance also play a key role in propagation. In particular, when specific parameters are selected, the CLED beam can form a transmission state with beam width and transverse pattern both invariant, i.e., CLED soliton forms. For general parameters, the transverse multi-peak patterns of the CLED beam evolutes periodically, i.e., a generalized higher-order soliton or breather forms.