Rogue waves in shallow water

Most of the processes resulting in the formation of unexpectedly high surface waves in deep water (such as dispersive and geometrical focusing, interactions with currents and internal waves, reflection from caustic areas, etc.) are active also in shallow areas. Only the mechanism of modulational instability is not active in finite depth conditions. Instead, wave amplification along certain coastal profiles and the drastic dependence of the run-up height on the incident wave shape may substantially contribute to the formation of rogue waves in the nearshore. A unique source of long-living rogue waves (that has no analogues in the deep ocean) is the nonlinear interaction of obliquely propagating solitary shallow-water waves and an equivalent mechanism of Mach reflection of waves from the coast. The characteristic features of these processes are (i) extreme amplification of the steepness of the wave fronts, (ii) change in the orientation of the largest wave crests compared with that of the counterparts and (iii) rapid displacement of the location of the extreme wave humps along the crests of the interacting waves. The presence of coasts raises a number of related questions such as the possibility of conversion of rogue waves into sneaker waves with extremely high run-up. Also, the reaction of bottom sediments and the entire coastal zone to the rogue waves may be drastic.     

Dust‐ion‐acoustic solitary waves in a magnetized dusty pair‐ion plasma with Cairns‐Gurevich electrons and opposite polarity dust particles

The nonlinear dust‐ion‐acoustic (DIA) solitary structures have been studied in a dusty plasma, including the Cairns‐Gurevich distribution for electrons, both negative and positive ions, and immobile opposite polarity dust grains. The external magnetic field directed along the z‐axis is considered. By using the standard reductive perturbation technique and the hydrodynamics model for the ion fluid, the modified Zakharov–Kuznetsov equation was derived for small but finite amplitude waves and was provided the solitary wave solution for the parameters relevant. Using the appropriate independent variable, we could find the modified Korteweg–de Vries equation. By plotting some figures, we have discussed and emphasized how the different plasma values, such as the trapping parameter, the positive (or negative) dust number density, the non‐thermal electron parameter, and the ion cyclotron frequency, can influence the solitary wave structures. In addition, using the bifurcation theory of planar dynamical systems, we have extracted the centre and saddle points and illustrated the phase portrait of such a system for some particular plasma parameters. Finally, we have graphically investigated the behaviour of the solitary energy wave by changing the plasma values as well as by calculating the instability criterion; we have also discussed the growth rate of the solitary waves. The results could be useful for studying the physical mechanism of nonlinear propagation of DIA solitary waves in laboratory and space plasmas where non‐thermal electrons, pair‐ions, and dust particles can exist.     

Growth of wave penetration under wind amplification

The penetration of nonlinear wind waves into the depth of a liquid is investigated in a laboratory channel. It is found that the amplitude of nonlinear waves on deep water depends only on the wave steepness normalized by the wave length. An expression is found that connects the growth of the wave mass with the wave steepness.     

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the combination of the third- and fifth-order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are close to solitons of the modified Korteweg-de Vries equation. However, the height of large-amplitude solutions has a limit approaching which solitary waves widen and acquire a table like shape similar to soluitons of the Gardner equation. Numerical calculations confirm that the collision of solitons of the derived equation is inelastic. Inelasticity is the most pronounced in the interaction of unipolar pulses. The direction of the shift of the phase of the higher-amplitude soliton owing to the interaction of solitons of different polarities depends on the amplitudes of the pulses.     

Non-dispersive traveling waves in inclined shallow water channels

Existence of traveling waves propagating without internal reflection in inclined water channels of arbitrary slope is demonstrated. It is shown that traveling non-monochromatic waves exist in both linear and nonlinear shallow water theories in the case of a uniformly inclined channel with a parabolic cross-section. The properties of these waves are studied. It is shown that linear traveling waves should have a sign-variable shape. The amplitude of linear traveling waves in a channel satisfies the same Green's law, which is usually derived from the energy flux conservation for smoothly inhomogeneous media. Amplitudes of nonlinear traveling waves deviate from the linear Green's law, and the behavior of positive and negative amplitudes are different. Negative amplitude grows faster than positive amplitude in shallow water. The phase of nonlinear waves (travel time) is described well by the linear WKB approach. It is shown that nonlinear traveling waves of any amplitude always break near the shoreline if the boundary condition of the full absorption is applied.     

内孤立波沿缓坡地形传播特性的实验研究

以南中国海东北部海域底部缓坡地形为背景, 在大型重力式分层流水槽中模拟了下凹型内孤立波沿缓坡地形传播过程中的浅化、破碎、分裂等现象, 利用分层染色标识方法和多点组合探头阵列技术对内孤立波沿缓坡地形演化特征进行了定性分析和定量测量. 实验表明: 浅化效应使内孤立波传播速度减小, 对大振幅内孤立波具有抑制作用, 对小振幅波具有放大效应; 浅化效应可导致内孤立波的剪切失稳及破碎, 还可导致大振幅内孤立波的分裂. 利用Miles稳定性理论可定性描述内孤立波沿缓坡地形传播时发生不稳定状态的位置, 实验结果与理论分析相符合. 关键词： 分层流 缓坡地形 内孤立波 不稳定性     

Local run-up amplification by resonant wave interactions

Until now, the analysis of long wave run-up on a plane beach has been focused on finding its maximum value, failing to capture the existence of resonant regimes. One-dimensional numerical simulations in the framework of the nonlinear shallow water equations are used to investigate the boundary value problem for plane and nontrivial beaches. Monochromatic waves, as well as virtual wave-gage recordings from real tsunami simulations, are used as forcing conditions to the boundary value problem. Resonant phenomena between the incident wavelength and the beach slope are found to occur, which result in enhanced run-up of nonleading waves. The evolution of energy reveals the existence of a quasiperiodic state for the case of sinusoidal waves. Dispersion is found to slightly reduce the value of maximum run-up but not to change the overall picture. Run-up amplification occurs for both leading elevation and depression waves.     

Interfacial waves with free-surface boundary conditions: an approach via a model equation

In a two-fluid system where the lower fluid is bounded below by a rigid bottom and the upper fluid is bounded above by a free surface, two kinds of solitary waves can propagate along the interface and the free surface: classical solitary waves characterized by a solitary pulse or generalized solitary waves with nondecaying oscillations in their tails in addition to the solitary pulse. The classical solitary waves move faster than the generalized solitary waves. The origin of the nonlocal solitary waves can be understood from a physical point of view. The dispersion relation for the above system shows that short waves can propagate at the same speed as a “slow” solitary wave. The interaction between the solitary wave and the short waves creates a nonlocal solitary wave. In this paper, the interfacial-wave problem is reduced to a system of ordinary differential equations by using a classical perturbation method, which takes into consideration the possible resonance between short waves and “slow” solitary waves. In the past, classical Korteweg–de Vries type models have been derived but cannot deal with the resonance. All solutions of the new system of model equations, including classical as well as generalized solitary waves, are constructed. The domain of validity of the model is discussed as well. It is also shown that fronts connecting two conjugate states cannot occur for “fast” waves. For “slow” waves, fronts exist but they have ripples in their tails.     

Intensification of three-dimensional waves by wind on deep water

It was found that the intensification of waves by wind includes consecutive cycles of growth of the longest three-dimensional wave’s steepness and its decay into even longer waves. The ratio of the lengths of waves that arise during decay was obtained as a function of the Froude number, and the steepness of the leading slope of the longest wave was obtained as a function of the wave’s steepness. An expression describing the depth of nonlinear waves’ penetration into a water column as a function of their steepness was found.     

Dynamics of radiation in optical fibers with a dispersive effective cross-section area of the mode

By using a graded-index optical fiber with a parabolic distribution of the refractive index as an example, it is shown that shock waves of envelopes can arise in optical fibers with a significantly dispersive (frequency-dependent) effective cross-section area of the mode. In principle, the shock wave caused by this dispersion can appear at the leading edge of the wave packet. The possibility of forming soliton pulses in media with a dispersive area of the mode is considered.     

Ion acoustic shock waves in a relativistic electron-positron-ion plasmas

The Korteweg-de Vries-Burger (KdVB) equation is derived for ion acoustic shock waves in a weakly relativistic electron-positron-ion plasma. Electrons, positrons are considered isothermal and ions are relativistic. The travelling wave solution has been acquired by employing the tangent hyperbolic method. The vivid display of the graphical results is presented and analyzed. It is observed that amplitude and steepness of the shock wave decrease with increase of the relativistic streaming factor, the positron concentration and they increase with the increase of the coefficient of kinematic viscosity and vice versa. It is determined that at low temperature the shock wave propagates, whereas at very high temperature the solitary wave propagates in the system. The results may have relevance in astrophysical plasmas as well as in inertial confinement fusion plasmas.     

Design and fabrication of emissive biased limiter and its effect on tokamak plasma

In this paper, an emissive-biased limiter (EBL) was designed and fabricated then the magneto hydrodynamic activity was investigated based on Mirnov oscillations and hard X-ray spectroscopy through the tokamak plasma biasing. The EBL is positioned at r/a=0.92, and the biased voltage, which is varied from?250 to 250 V, applied between the head of the emissive limiter and vacuum chamber. Furthermore, the effects of the biased limiter for both negative and positive applied voltages are measured, and the results are compared with cold-biased limiter. As the results of IR-T1 tokamak suggest, in emissive negative polarity, the duration of plasma current is increased, compared with no bias, cold positive and emissive positive polarities. The amplitude of Mirnov fluctuations in emissive negative polarity is larger and more regular, compared with emissive positive polarity. The amplitude of the hard X-ray fluctuation in emissive negative polarity is very low compared with cold negative, cold positive and emissive positive polarities which leads to minimum energy loss.     

Propagation of short pulses in a medium with frequency dispersion

Propagation of short pulses of electromagnetic radiation in a transparent medium with a frequency dispersion is considered for the linear and nonlinear modes. The role of back reflection of radiation in the spectral regions with the negative dielectric constant was clarified. For the nonlinear mode, the exact form of the stationary solitary wave whose profile includes a point with infinite steepness is found.     

Dynamics of Periodic Waves in Bose-Einstein Condensate with Time-Dependent Atomic Scattering Length

Evolution of periodic waves and solitary waves in Bose-Einstein condensates （BECs） with time-dependent atomic scattering length in an expulsive parabolic potential is studied. Based on the mapping deformation method, we successfully obtain periodic wave solutions and solitary wave solutions, including the bright and dark soliton solutions.The results in this paper include some in the literatures [Phys. Rev. Lett. 94 （2005） 050402 and Chin. Phys. Left. 22 （2005） 1855].     

Propagation features of head-on collision dust acoustic solitary waves in four-component quantum plasmas

ABSTRACT

In framework of the extended Poincaré–Lighthill–Kuo, the properties of dust acoustic (DA) solitary wave’s interaction are investigated in four-component quantum dusty plasma. Two Korteweg–de Vries equations describing the colliding DA solitary waves are derived by eliminating the secularities. By knowing the explicit form of the solitary wave solutions, the leading phase changes, trajectories and phase shifts are obtained, accordingly. The effects of various physical parameters such as the quantum mechanical parameters, the charge ratio between positive and negative dust particles, the mass ratio between negative and positive dust particles and the ratio of electron to ion temperatures are studied extensively. Our findings showed that these parameters play a significant role on the characteristics and basic features of DA solitary waves such as phase shifts in trajectories due to collision. The obtained results may be beneficial to understand well the collision of DA solitary waves that may occur in laboratory plasmas, space plasma as well as in plasma applications.     

Aspect of K-dV Solitons Studied with Different Stretching Co-ordinates

The present study focusses on the conspicuous properties of the temporal and spatial K-dV solitary waves obtainable by the use of different stretching co-ordinates. Singularities are observed in solitary waves which are due to the presence of negative ions and multiple electron-temperatures in the plasmas. It is discussed how those situations are tackled by the derivation of modified K-dV equations to describe the evolution of propagating solitons. Moreover, the soliton profiles admit the interactions of negative ions and multiple electron-temperatures, which are playing the role of creating the steepness in soliton profiles. Simultaneously more solitary waves follow therein resembling the evolution of experimentally observed soliton characteristics and finally indicating the possible interest to put forth for experimental plasma investigations.     

Effects of axial magnetic field and nonextensivity on small amplitude dust acoustic solitary waves in dusty plasma

The nonlinear propagation of small amplitude dust‐acoustic (DA) solitary waves in magnetized dusty plasma consisting of negatively charged mobile dust fluid, and Boltzmann‐distributed electrons and ions with two distinct temperatures following a q‐nonextensive distribution are investigated. In this article, a number of nonlinear equations, namely, the Korteweg–de‐Vries (K‐dV) equations, have been derived by employing the reductive perturbation technique that is valid for a small but finite amplitude limit. The effects of nonextensivity of ions with two distinct temperatures and dust concentration on the amplitude and width of DA solitary waves are investigated theoretically. It is observed that both the nonextensive and low‐temperatures ions significantly modify the basic properties and polarities of DA solitary waves. It is shown that both positive and negative potential DA solitons occur in this case. The implications of these results to some astrophysical environments and space plasmas (e.g., stellar polytropes, peculiar velocity distributions of galaxies, and collisionless thermal plasma), and laboratory dusty plasma systems are briefly mentioned.     

Solitary Wave Solutions of a Generalized Derivative Nonlinear Schrödinger Equation

With the aid of a class of nonlinear ordinary differential equation (ODE) and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Schrödinger equation (GDNLSE) have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.     

Multi-dimensional instability of electrostatic solitary waves in a warm multi-ion dust plasma

Study of dust ion acoustic waves in a magnetized dusty plasmas composed of negatively or positively charged static dust, positive and negative ions, as well as kappa distribution electrons is presented. The Zakharov–Kuznetsov (ZK) equation is derived via reductive perturbation technique. The solitary wave solution of ZK equation is given and the multi-dimensional instability of these solitary waves is investigated via small k perturbation method. The instability criterion and growth rate relying on obliqueness, superthermality, positive ion thermal pressure, relative ion number density, magnetic field strength, and direction cosines are discussed for five cases. The results are beneficial to understand different nonlinear characteristics of unstable electrostatic disturbances in laboratory and space plasmas.     

Evolution of a langmuir wave in a weakly inhomogeneous plasma with a positive concentration gradient

Spatial evolution of a Langmuir wave excited by external sources in a weakly inhomogeneous electron plasma without external sources is considered for a small positive gradient of the plasma concentration in the direction of propagation of the wave. At the first state of the evolution, the dispersion of the wave is close to linear. When the phase velocity is doubled, the second stage of the evolution begins. The wave loses its individuality and becomes a hybrid of two waves. Its profile acquires the shape of an alternating sequence of fragments of these waves. The wave dispersion is determined by the dispersion of each fragment. In the course of evolution, the spacing between the equilibrium values of the wave fragments increases; as a result, the wave decays into two waves, which are also loaded by trapped electrons. Prior to decay, the humps of the wave become steeper; as a result, at the instant of the decay, the wave is transformed into a sequence of solitons with different polarities.