Hybrid solutions to Mel’nikov system

Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.

     

Double degeneration on second-order breather solutions of Maxwell–Bloch equation

The breather solutions of the Maxwell–Bloch equations in a two-level resonant system associated with the self-induced transparency phenomenon are constructed by the Darboux transformation. After constructing the formulas of the second-order breather solutions, the double degeneration and hybrid solutions are studied by the analytical form as well as figures. Our results might be helpful in such application or prevention of the rogue waves from breather solution interactions and degeneration in the nonlinear optical systems associated with the Maxwell–Bloch equations.     

Discrete breathers and modulational instability in a discrete $$\varvec{\phi ^{4}}$$ nonlinear lattice with next-nearest-neighbor couplings

The properties of discrete breathers and modulational instability in a discrete \(\phi ^{4}\) nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the \(q=\pi \) plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.     

Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium

A (3+1)-dimensional coupled nonlinear Schrödinger equation with different inhomogeneous diffractions and dispersion is investigated, and rogue wave and combined breather solutions are constructed. Different diffractions and dispersion of medium lead to the repeatedly excited behaviors of rogue wave and combined breather in the dispersion/diffraction decreasing system. These repeated behaviors including complete excitation, rear excitation, peak excitation and initial excitation are discussed.     

Bilinear forms and soliton interactions for two generalized KdV equations for nonlinear waves

Korteweg–de Vries (KdV)-type equations can describe the nonlinear waves in fluids, plasmas, etc. In this paper, two generalized KdV equations are under investigation. Bilinear forms of which are constructed with the Bell polynomials and an auxiliary variable. \(N\) -soliton solutions are given through the Hirota direct method. Via the asymptotic analysis, the soliton interactions of the first generalized KdV equation are analyzed, which turn out to be elastic. Singular breather solutions have been derived from the two-soliton solutions. The collision between soliton and singular breather appears to be elastic, and the bound states of soliton and singular breather are exhibited. Unlike the first one, the other generalized KdV equation can only support the bound states of solitons, for the regular and singular solitons alike.     

Resonant solutions and breathers to the BKP equation

In this paper, we derive resonant and breather solutions from multi-soliton solutions of the B-type Kadomtsev–Petviashvili (BKP) equation of fourth order via the Hirota bilinear method. We first discuss N-soliton solutions of the BKP equation and use the linear superposition principle to generate N-resonant solutions. Subsequently, we construct complexiton and breather solutions and finally, study the dynamics of some selected solutions with the aid of 3D plots, contour plots and density plots.

     

初始呼吸子对Sine-Gordon系统动力行为的影响

研究了简谐外力扰动下,计入Peierls-Nabarro力和固体粘性效应,在Neumarm边界条件下杆的运动,这个运动可以用Sine-Gordon型方程来模拟．通过有限差分将无穷维的Sine-Gordon型系统近似为有限维系统．当外力的幅值和初始条件中呼吸子的相位发生变化时,系统的解呈现出丰富的空间结构和时间行为,对于一定的外力幅值和相位,系统具有混沌解．发现初始呼吸子的相位也是影响系统运动状态的重要因素。     

Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation

Nonlinear Dynamics - Based on Darboux transformation, we generate breather solutions sitting on zero background via module resonance for complex modified KdV equation. Then, some novel soliton...     

A ship motion forecasting approach based on empirical mode decomposition method hybrid deep learning network and quantum butterfly optimization algorithm

Studies of the shallow water waves are active, possessing the applications in ocean engineering, marine environment, atmospheric science, etc. In this paper, we investigate a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Hirota method and symbolic computation help us work out (1) a bilinear form, (2) N-soliton solutions with N being a positive integer, (3) the higher-order breather solutions, (4) periodic-wave solutions and (5) hybrid solutions composed of one first-order breather and one soliton/two solitons. Moreover, we provide some nonlinear phenomena described by the associated solutions. All of the obtained results are determined via the time-dependent coefficients of that equation.

     

Interactions of the vector breathers for the coupled Hirota system with $$4\times 4$$ 4 × 4 Lax pair

Nonlinear Dynamics - In this paper, we investigate the interactions of the vector breathers for the coupled Hirota system with $$4\times 4$$ Lax pair. Firstly, we give the first-order breather...     

New mixed solutions generated by velocity resonance in the $$(2+1)$$ ( 2 + 1 ) -dimensional Sawada–Kotera equation

Nonlinear Dynamics - Based on the N-soliton solutions of the $$(2+1)$$ -dimensional Sawada–Kotera equation, the collisions among lump waves, line waves, and breather waves are studied in this...     

New no-traveling wave solutions for the Liouville equation by Bäcklund transformation method

With the aid of the known Bäcklund transformation, starting from some given traveling solutions, we consider new exact no-traveling wave solutions to the Liouville equation, and a series of breather soliton solutions, doubly periodic solutions, two-soliton solutions as well as periodic-soliton solutions are obtained.     

Study of Two-Dimensional Axisymmetric Breathers Using Padé Approximants

We analyze axisymmetric, spatially localized standing wave solutions with periodic time dependence (breathers) of a nonlinear partial differential equation. This equation is derived in the 'continuum approximation' of the equations of motion governing the anti-phase vibrations of a two-dimensional array of weakly coupled nonlinear oscillators. Following an asymptotic analysis, the leading order approximation of the spatial distribution of the breather is shown to be governed by a two-dimensional nonlinear Schrödinger (NLS) equation with cubic nonlinearities. The homoclinic orbit of the NLS equation is analytically approximated by constructing [2N × 2N] Padé approximants, expressing the Padé coefficients in terms of an initial amplitude condition, and imposing a necessary and sufficient condition to ensure decay of the Padé approximations as the independent variable (radius) tends to infinity. In addition, a convergence study is performed to eliminate 'spurious' solutions of the problem. Computation of this homoclinic orbit enables the analytic approximation of the breather solution.     

Resonance and deflection of multi-soliton to the (2+1)-dimensional Kadomtsev–Petviashvili equation

In this work, a modified three-soliton method with a perturbation parameter is proposed, and it is applied to the (2+1)-dimensional Kadomtsev–Petviashvili equation (KP), and new breather multi-soliton solutions are obtained. The dependence of new mechanical structures on the perturbed parameter for multi-soliton including resonance and deflection for KP equation are investigated and exhibited.     

Diffraction Managed Solitons with Zero Mean Diffraction

The problem of existence of soliton solutions in a diffraction managed NLS is considered in the case of zero mean diffraction. Existence of localized breather solutions for the averaged equation is shown using Ekeland’s variational principle in the corresponding minimization of the Hamiltonian procedure. The corresponding minimizer is shown to be spatially well-localized and orbitally stable.      

“Leapfrogging” solitons in a system of coupled KdV equations

A system of two KdV equations coupled by small linear dispersive terms is considered. This system describes, for example, resonant interaction of two transverse gravity internal wave modes in a shallow stratified liquid. In the framework of an approach based on Hamilton's equations of motion, evolution equations for parameters of two solitons belonging to different wave modes are obtained in the adiabatic approximation. It is demonstrated that when the solitons' velocities are sufficiently lose, the solitons may form a breather-like oscillatory bound state, which provides a natural explanation for recent numerical experiments demonstrating “leapfrogging” motion of the two solitons. The frequency and the maximum amplitude of the “breather”'s internal oscillations are obtained. For the case when the relative velocity of the solitons is not small, perturbation-induced phase shifts of the two colliding free solitons are calculated. Then emission of radiation (small-amplitude quasilinear waves) by an oscillating “breather,” also detected in the numerical experiments, is investigated in the framework of the perturbation theory based on the inverse scattering transform. The intensity of the emission is calculated. Radiative effects accompanying collision of the free solitons are also investigated.     

On the characterization of breather and rogue wave solutions and modulation instability of a coupled generalized nonlinear Schrödinger equations

We construct Darboux transformation of a coupled generalized nonlinear Schrödinger (CGNLS) equations and obtain exact analytic expressions of breather and rogue wave solutions. We also formulate the conditions for isolating these solutions. We show that the rogue wave solution can be found only when the four wave mixing parameter becomes real. We also investigate the modulation instability of the steady state solution of CGNLS system and demonstrate that it can occur only when the four wave mixing parameter becomes real. Our results give an evidence for the connection between the occurrence of rogue wave solution and the modulation instability.     

Space–time evolution of optical breathers and modulation instability patterns in metamaterial waveguides

We present numerical and theoretical investigations of the spontaneous emergence of noise-driven modulation instability patterns in a metamaterial waveguide, which involves the generation of optical breather waves such as the Peregrine soliton, Akhmediev breathers and Kuznetsov–Ma breathers. We show that the intrinsic properties of the metamaterial waveguide, e.g. self-steepening and the magnetooptic effects, offer the potential to control the formation and subsequent spectral and temporal dynamics of these localized nonlinear waves. Such internal or external perturbations break the symmetry of the spectrum of nonlinear waves, thus leading to the existence of a controllable characteristic group velocity in their space–time evolution.     

Weakly nonlocal envelope solitary waves: numerical calculations for the Klein-Gordon (φ) equation

“Weakly nonlocal” solitary waves differ from ordinary solitary waves by possessing small amplitude, oscillatory “wings” that extend indefinitely from the large amplitude “core”. Such generalized solitary waves have been discovered in capillarygravity water waves, particle physics models, and geophysical Rossby waves. In this work, we present explicit calculations of weakly nonlocal envelope solitary waves. Each is a sine wave modulated by a slowly-varying “envelope” that itself propagates at the group velocity. Our example is the cubically nonlinear Klein-Gordon equation, which is a model in particle physics (φ⁴ theory) and in electrical engineering (with a different sign). Both cases have weakly nonlocal“breather” solitons. Via the Lorentz invariance, each breather generates a one-parameter family of nonlocal envelope solitary waves. The φ⁴ breather was described and calculated in earlier work. This generates envelope solitons which have “wings” that are (mostly) proportional to the second harmonic of the sinusoidal factor. In this article, we calculate breathers and envelope solitary waves for the second, electrical engineering case. Since these, unlike the φ⁴ waves, contain only odd harmonics, the envelope solitary waves are nonlocal only via the third harmonic.