Stability of hydraulic fall and sub-critical cnoidal waves in water flows over a bump

A forced Korteweg–de Vries (fKdV) equation can be used to model the surface wave of a two-dimensional water flow over a bump when the upstream Froude number is near one. The fKdV model typically has four types of solutions: sub-critical cnoidal waves, sub-critical hydraulic fall, transcritical upstream soliton radiation, and supercritical multiple solitary waves. This paper provides a numerical demonstration of the stability of the hydraulic falls and cnoidal waves solutions.     

A model equation for steady surface waves over a bump

The objective of this paper is to study the solutions of a model equation for steady surface waves on an ideal fluid over a semicircular or semielliptical bump. For upstream Froude number F>1, we show that the numerical solution of the equation has two branches and there is a cut-off value of F below which no solution exists. For F<1, the problem is reformulated to overcome the so-called infinite-mass dilemma. A branch of solutions and a cut-off value of F, above which no solution exists, are found. Furthermore, we also obtain a branch of hydraulic-fall solutions which decrease monotonically from upstream to downstream.     

Double sine-Gordon solitons in nematic liquid crystals under applied electric and magnetic fields

In this paper, we derive the dynamic equation of molecular motion for a twisted nematic liquid crystal (NLC) under applied electric and magnetic fields, and show that it takes the form of a double sine-Gordon (DSG) equation. Two kink and anti-kink solitary solutions of the liquid crystal molecules are obtained by using the F-expansion method to solve the DSG equation. Finally, we confirm that the twist of the NLC molecules can propagate in the form of solitary waves. The propagation velocity and amplitude of the kink and anti-kink solitons induced by the electric and magnetic fields are discussed.     

Gardner方程的孤立波解

用动力系统分支理论研究了Gardner方程,给出了分支参数空间以及许多孤立波解,扭子波解,在各种参数条件下,得到了所有显式的有界的精确的孤立波解和扭子波解。     

Grey and black optical solitary waves,and modulation instability analysis to the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

This paper addresses the nonlinear Schrödinger equation (NLSE) with Kerr law nonlinearity and perturbation terms in optical fibre. A class of grey and black optical solitary wave solutions of this equation are retrieved by adopting an appropriate solitary wave ansatz solution. These types of solitary waves play a vital role in understanding various physical phenomena in nonlinear systems. This lead to a constraint condition on the solitary wave parameters which must hold for the solitary waves to exist. Moreover, the modulation instability (MI) analysis of the model is studied by employing the concept of linear-stability analysis (LSA) and the MI gain spectrum is got. Physical interpretations of the acquired results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviours of the equation.     

On periodic and solitary pure gravity waves in water of infinite depth

Non-linear two-dimensional waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow irrotational. Gravity is taken into account, but surface tension is neglected. Accurate solutions are computed by boundary integral equation methods. Previous results on irregular waves are confirmed and extended. The computations strongly suggest the existence of non-periodic waves. These waves resemble generalised solitary waves in the sense that they are characterised by a train of oscillations in the far field.     

Nonlinear waves in an active-dissipative disperse medium

Nonlinear waves in a medium involving dissipation, dispersion, and enhancement described by the generalized Kuramoto-Sivashinsky equation are discussed. Analytical solutions of the equation are obtained in the form of solitary waves. For numerical modeling of the nonlinear waves a difference scheme is suggested. Interaction of nonlinear waves described by the Kuramoto-Sivashinsky model is considered. It is shown that for specified values of the problem parameters there is one solitary wave described by the initial model. The dependences of the velocity and amplitude of this wave on the problem parameters are determined. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 1, pp. 149–154, January–February, 1998.     

Fully nonlinear gravity-capillary solitary waves in a two-fluid system of finite depth

Large-amplitude waves at the interface between two laminar immisible inviscid streams of different densities and velocites, bounded together in a straight infinite channel are studied, when surface tension and gravity are both present. A long-wave approximation is used to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across it. Traveling waves of permanent form are studied and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves can be expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9, where 2h is the channel thickness. In the absence of gravity solitary waves are not possible but periodic ones are. Numerically constructed solitary waves are given for representative physical parameters.     

Nonlinear waves in the Pitaevskii-Gross equation

Nonlinear waves, solitary and periodic, are studied exactly in the Pitaevskii-Gross equation for the wave function of the condensate of a superfluid. We also study the relationship between these two waves and Bogoliubov's phonon, and the energies associated with these waves. The creation energy of a solitary wave with amplitudeA is proportional toA ^3/2. Solitary waves show interesting behavior on their collision due to their localized character. The effect of collision on solitary waves can be described by the phase shift. We give a formula of the phase shift on a collision of two solitary waves. We further discuss the decay of an arbitrary initial disturbance into solitary waves.On leave from Department of Physics, Kyushu University, Fukuoka, Japan.     

Two-dimensional generalized solitary waves and periodic waves under an ice sheet

Two-dimensional gravity waves travelling under an ice sheet are studied. The flow is assumed to be potential. Weakly nonlinear solutions are derived and fully nonlinear solutions are calculated numerically. Periodic waves and generalized solitary waves are studied.     

On pseudo-uniform flow in open channel hydraulics

Summary The differential equation that governs the longitudinal variation of the surface profileh(x) in steady plane channel flow is qualitatively discussed for the case of pseudo-uniform flow states. The solutions are either of the cnoidal or solitary wave type. It is shown that, among all cnoidal waves, solitary waves have minimum energy head. Further, surface profiles mustbreak whenever the Froude-number exceeds the value (which is close to experimentally determined values). The model equations are applied to the undular hydraulic jump, and it is shown how the equations can be used in practical situations of open channel flow hydraulics.With 11 Figures     

Non-existence of solitary water waves in three dimensions

In the subject of free-surface water waves, solitary waves play an important role in the theory of two-dimensional fluid motions. These are steady solutions to the Euler equations that are localized, positively elevated above the mean fluid level and travelling at velocities with supercritical Froude number. They provide a stable mechanism in bodies of water for transport of mass, momentum and energy over long distances. In this paper, we prove that in the three- (or higher-) dimensional problem of surface water waves, there do not exist any localized steady positive solutions to the Euler equations.     

Construction of stationary waves on a falling film

Waves on a film falling down a vertical wall exhibit many distinct features. They tend to be locally stationary over several wavelengths, viz. they travel with constant speeds and shapes over a long distance. In the limit of very long (solitary) waves, these stationary waves also exhibit two length scales with small and short capillary waves running ahead of a large tear-drop shaped hump. We present a spectral-element method for this difficult multi-scale free surface problem. A boundary layer approximation of the equation of motion allows a Fourier expansion in the streamwise direction in conjunction with a domain decomposition in the direction normal to the wall that eliminates numerical instability. This mixed method hence enjoys both the exponential convergence rate of a spectral technique and the numerical advantage provided by a compactly supported basis which yields sparse projected differential operators. All stationary wave families, parameterized by the wavelength, are then constructed using a Newton continuation scheme. The constructed waves are favorably compared to experimentally measured wave shapes.     

Role of higher-order scattering in solutions to the forward and inverse optical-imaging problems in random media

From analytical and numerical solutions that predict the scattering of diffuse photon density waves and from experimental measurements of changes in phase shift theta and ac amplitude demodulation M caused by the presence of single and double cylindrical heterogeneities, we show that second- and higher-order perturbations can affect the prediction of the propagation characteristics of diffuse photon density waves. Our experimental results for perfect absorbers in a lossless medium suggest that the performance of fast inverse-imaging algorithms that use first-order Born or Rytov approximations might have inherent limitations compared with inverse solutions that use iterative solutions of a linear perturbation equation or numerical solutions of the diffusion equation.     

Vector vortex solitons in two-component Bose–Einstein condensates with modulated nonlinearities and a harmonic trap

We introduce vector solitary waves in two-component Bose–Einstein condensates with spatially modulated nonlinearity coefficients and a harmonic trapping potential. Using the self-similarity method, novel vector solitary waves are built with the help of Whittaker function, including multipole solutions and necklace rings. The stability of vortex soliton pairs is examined by direct numerical simulation; the results show that a new class of stable low-order vortex soliton pairs with n = 2 and m ≤ 3 can be supported by the spatially modulated interaction in the harmonic trap. Higher order vector-vortex soliton is found unstable over prolonged distances.     

Applicability of the method of fundamental solutions to 3-D wave–body interaction with fully nonlinear free surface

A numerical model for three-dimensional fully nonlinear free-surface waves is developed by applying a boundary-type meshless approach with a leap-frog time-marching scheme. Adopting Gaussian Radial Basis Functions to fit the free surface, a non-iterative approach to discretize the nonlinear free-surface boundary is formulated. Using the fundamental solutions of the Laplace equation as the solution form of the velocity potential, free-surface wave problems can be solved by collocations at only a few boundary points since the governing equation is automatically satisfied. The accuracy of the present method is verified by comparing the simulated propagation of a solitary wave with an exact solution. The applicability of the present model is illustrated by applying it to the problem of a solitary wave running up on a vertical surface-piercing cylinder and the problem of wave generation in infinite water depth by a submerged moving object.     

A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves

The existence of a line solitary-wave solution to the water-wave problem with strong surface-tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg and G. de Vries and rigorously confirmed a century later by C. J. Amick and K. Kirchg?ssner in 1989. A model equation derived by B. B. Kadomtsev and V. I. Petviashvili in 1970 suggests that the Korteweg-de Vries line solitary wave belongs to a family of periodically modulated solitary waves which have a solitary-wave profile in the direction of motion and are periodic in the transverse direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. It is shown that the Korteweg-de Vries solitary wave undergoes a dimension-breaking bifurcation that generates a family of periodically modulated solitary waves. The term dimension-breaking phenomenon describes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.     

非线性尘埃等离子体孤立子波变分迭代解

在全球气候变暖的极端反常的情形下，大气尘埃的扩散现象会带来巨大的灾害．本文研究了大气尘埃等离子体扩散的一类广义非线性孤立子波模型．首先对非扰动情形下利用待定系数法得到孤立子波解的解析表示式．其次用广义变分迭代的方法求出对应的变分乘子并构造变分迭代式，依次求出孤子波的各次迭代解．然后用行波变换得到广义非线性尘埃等离子体扰动模型的孤立子波的各次近似解．最后，由得到解的近似函数序列据变分理论知，在自变量的一定区域内此序列为一致收敛的．因此便证明了迭代解的极限函数是尘埃等离子体低频振动非线性方程的精确解．本文得到的近似解是尘埃等离子体的低频振动孤立子波的近似解析解，据它可用解析运算来求出相关量的物理性态，如孤立子波的波峰值．可以根据本文理论采取相应措施，避免出现电荷超高密度的聚集而导致放电击穿现象等．     

Hamiltonian formulation for solitary waves propagating on a variable background

Solitary waves propagating on a variable background are conventionally described by the variable-coefficient Korteweg-de Vries equation. However, the underlying physical system is often Hamiltonian, with a conserved energy functional. Recent studies for water waves and interfacial waves have shown that an alternative approach to deriving an appropriate evolution equation, which asymptotically approximates the Hamiltonian, leads to an alternative variable-coefficient Korteweg-de Vries equation, which conserves the underlying Hamiltonian structure more explicitly. This paper examines the relationship between these two evolution equations, which are asymptotically equivalent, by first discussing the conservation laws for each equation, and then constructing asymptotically a slowly-varying solitary wave.     

Turbulence and Coherent Structures in Two-Component Bose Condensates

We elucidate the self-evolution of two Bose gases from a strongly non - equilibrium initial state. Large scale numerical simulations of the coupled nonlinear Schr?dinger (NLS) equations are used to follow the evolution of the system from weak turbulence to strong turbulence to superfluid turbulence in the long-wavelength region of energy space with a formation of a tangle of topological defects. The addition of the second gas increases the number of condensed particles in the first gas. It is shown that the large wavelength part of the fields evolves into coherent structures identified as solitary wave complexes of the coupled nonlinear Schr?dinger equations. The families of the solitary waves moving on uniform background or along the topological defects are obtained as solutions of the coupled Gross–Pitaevskii equations. It is shown that there exist three continuous families of such solutions with or without the cusp in energy-momentum space.