Could rogue waves be used as efficient weapons against enemy ships?

The title of this paper is more provocative than serious, since its aim, in this special issue of EPJ, is mainly to heat up discussions and debates on the topic of rogue waves. Our goal is to reach a better understanding of the phenomenon, rather than to use it as a destructive tool. It is clear from previous studies that rogue waves are formed due to at least two mechanisms of amplification, rather than in a single stage. The first mechanism is modulation instability and Akhmediev breathers while the second one is multiple soliton collisions. In this short article, we consider soliton collisions with energy exchange as one of the important mechanisms of nonlinear amplification that can irreversibly lead to the creation of giant waves that we can call “rogue waves”.     

Are there waves in elastic wave turbulence?

An thin elastic steel plate is excited with a vibrator and its local velocity displays a turbulentlike Fourier spectrum. This system is believed to develop elastic wave turbulence. We analyze here the motion of the plate with a two-point measurement in order to check, in our real system, a few hypotheses required for the Zakharov theory of weak turbulence to apply. We show that the motion of the plate is indeed a superposition of bending waves following the theoretical dispersion relation of the linear wave equation. The nonlinearities seem to efficiently break the coherence of the waves so that no modal structure is observed. Several hypotheses of the weak turbulence theory seem to be verified, but nevertheless the theoretical predictions for the wave spectrum are not verified experimentally.     

Dynamics of nonautonomous rogue waves in Bose–Einstein condensate

We study rogue waves of Bose–Einstein condensate (BEC) analytically in a time-dependent harmonic trap with a complex potential. Properties of the nonautonomous rogue waves are investigated analytically. It is reported that there are possibilities to ‘catch’ rogue waves through manipulating nonlinear interaction properly. The results provide many possibilities to manipulate rogue waves experimentally in a BEC system.     

Formation mechanism of asymmetric breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations

Based on the developed Darboux transformation, we investigate the exact asymmetric solutions of breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations. As an example, some types of exact breather solutions are given analytically by adjusting the parameters. Moreover, the interesting fundamental problem is to clarify the formation mechanism of asymmetry breather solutions and how the particle number and energy exchange between the background and soliton ultimately form the breather solutions. Our results also show that the formation mechanism from breather to rogue wave arises from the transformation from the periodic total exchange into the temporal local property.     

Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation

We present an explicit analytical form of first and second order rogue waves for distributive nonlinear Schrödinger equation (NLSE) by mapping it to standard NLSE through similarity transformation. Upon obtaining the rogue wave solutions, we study the propagation of rogue waves through a periodically distributed system for the two cases when Wronskian of dispersion and nonlinearity is (i) zero, (ii) not equal to zero. For the former case, we discuss a mechanism to control their propagation and for the latter case we depict the interesting features of rogue waves as they propagate through dispersion increasing and decreasing fiber.     

Modulation instability,rogue waves and conservation laws in higher-order nonlinear Schrodinger equation

In this paper,the modulation instability(MI),rogue waves(RWs)and conseryation laws of the coupled higher-order nonlinear Schrodinger equation are investigated.According to MI and the 2×2 Lax pair,Darboux-dressing transformation with an asymptotic expansion method,the existence and properties of the one-,second-,and third-order RWs for the higher-order nonlinear Schrodinger equation are constructed.In addition,the main characteristics of these solutions are discussed through some graphics,which are draw widespread attention in a variety of complex systems such as optics,Bose-Einstein condensates,capillary fow,superfluidity,fluid dynamics,and finance.In addition,infinitely-many conservation laws are established.     

Heavy ion–acoustic rogue waves in magnetized electron–positron multi-ion plasmas

Non-linear heavy ion-acoustic waves (HIAWs) are studied in a homogeneous magnetized four-component multi-ion plasma composed of inertial heavy negative ions, light positive ions, and inertia-less non-extensive electrons and positrons. The non-linear Schrödinger equation is derived in this model using the perturbation method. The criteria for modulational instability of HIAWs and the basic features of finite-amplitude heavy ion acoustic rogue waves (HIARWs) are investigated. The presence of the magnetic field was found to reduce the amplitude of HIARWs and enhances the stability. It is interesting to note that increasing positive ion mass causes decreases in the amplitude and width of rogue waves, which is opposite behaviour to that demonstrated in the previous study of these waves in an unmagnetized plasma. Furthermore, it is also shown that striking parameters, such as the non-extensive parameter, the positron number density, the electron number density, the electron temperature, and the magnetic field parameter, play an undeniable role on the stability of waves packets. The findings of the present investigation may be of wide relevance to some plasma environments, such as active galactic nuclei, pulsar magnetospheres, and other magnetic confinement systems.     

Symmetry and asymmetry rogue waves in two-component coupled nonlinear Schrdinger equations

By means of the modified Darboux transformation we obtain some types of rogue waves in two-coupled nonlinear Schr ¨odinger equations.Our results show that the two components admits the symmetry and asymmetry rogue wave solutions,which arises from the joint action of self-phase,cross-phase modulation,and coherent coupling term.We also obtain the analytical transformation from the initial seed solution to unique rogue waves with the bountiful pair structure.In a special case,the asymmetry rogue wave can own the spatial and temporal symmetry gradually,which is controlled by one parameter.It is worth pointing out that the rogue wave of two components can share the temporal inversion symmetry.     

Can ion-acoustic waves in plasma be backward waves?

The dispersion relation for ion-acoustic waves in plasma with ion flow has been analyzed. It is shown that these waves may exist (under certain conditions) in the form of backward waves with antiparallel group and phase velocities. The range of ion flow velocities allowing implementation of backward ion-acoustic waves is found.     

Characteristics of the solitary waves and rogue waves with interaction phenomena in a (2 + 1)-dimensional Breaking Soliton equation

Under inquisition in this paper is a $(2+1)$-dimensional Breaking Soliton equation, which can describe various nonlinear scenarios in fluid dynamics. Using the Bell polynomials, some proficient auxiliary functions are offered to apparently construct its bilinear form and corresponding soliton solutions which are different from the previous literatures. Moreover, a direct method is used to construct its rogue wave and solitary wave solutions using particular auxiliary function with the assist of bilinear formalism. Finally, the interactions between solitary waves and rogue waves are offered with a complete derivation. These results enhance the variety of the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering.     

Analytical solutions and rogue waves in （3＋1）-dimensional nonlinear SchrSdinger equation

Analytical solutions in terms of rational-like functions are presented for a(3+1)-dimensional nonlinear Schrdinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz.Several free functions of time t are involved to generate abundant wave structures.Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg–de Vries equation

Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.     

Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa–Satsuma case

We present the lowest order rogue wave solution of the Sasa–Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of the Peregrine solution appears when the extension parameter of the SSE is reduced to zero.     

Control of spiral waves and turbulent states in a cardiac model by travelling—wave perturbations

We propose a travelling-wave perturbation method to control the spatiotemporal dynamics in a cardiac model.It is numerically demonstrated that the method can successfully suppress the wave instability(alternans in action potential duration) in the one-dimensional case and converty spiral waves and turbulent states to the normal travelling wave states in the two-dimensional case.An experimental scheme is suggested which may provide a new design for a cardiac defibrillator.     

Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell–Bloch equations

Under investigation in this paper are the inhomogeneous nonlinear Schrödinger Maxwell–Bloch (INLS-MB) equations which model the propagation of optical waves in an inhomogeneous nonlinear light guide doped with two-level resonant atoms. Higher-order nonautonomous breather as well as rogue wave solutions in terms of the determinants for the INLS-MB equations are presented via the n$n$-fold variable-coefficient modified Darboux transformation. The interactions among two nonautonomous breathers are graphically discussed, including the fundamental breather, bound breather, two-breather compression and two-breather evolution, etc. Moreover, several patterns of the higher-order rogue waves are also exhibited, such as the square rogue wave, two- and three-order periodic rogue waves, periodic fission and fusion, two-order stationary rogue waves, and recurrence of the two-order rogue waves. The character of the trajectory of the two-order periodic rogue wave is analyzed. Additionally, a novel type of interaction, namely, the collision between the breather and long-lived rogue waves, is found to be elastic. Our results could be useful for controlling the nonautonomous optical breathers and rogue waves in the inhomogeneous erbium doped fiber.     

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.     

Primordial perturbations and non-gaussianities in Ho?ava-Lifshitz gravity

We investigate primordial perturbations and non-gaussianities in the Ho?ava-Lifshitz theory of gravitation. In the UV limit, the scalar perturbation in the Ho?ava theory is naturally scale-invariant, ignoring the details of the expansion of the Universe. One may thus relax the exponential inflation and the slow-roll conditions for the inflaton field. As a result, it is possible that the primordial non-gaussianities, which are " slow-roll suppressed” in the standard scenarios, become large. We calculate the non-gaussianities from the bispectrum of the perturbation and find that the equilateral-type non-gaussianity is of the order of unity, while the local-type non-gaussianity remains small, as in the usual single-field slow-roll inflation model in general relativity. Our result is a new constraint on Ho?ava-Lifshitz gravity.