Fully resonant soliton interactions in the Whitham–Broer–Kaup system based on the double Wronskian solutions

With the aim of exploring whether the (1+1)-dimensional coupled nonlinear evolution equations admit abundant soliton interactions, like the cases in the Kadomtsev–Petviashvili II equation, we in this paper study the double Wronskian solutions to the Whitham–Broer–Kaup (WBK) system. We give the parametric condition for two double Wronskians to generate the non-singular, non-trivial and irreducible soliton solutions. Via the asymptotic analysis of two double Wronskians, we show that the soliton solutions of the WBK system is in general linearly combined of fully resonant (M,N)- and (M?1,N+1)-soliton configurations. It turns out that the WBK system can exhibit various complex soliton structures which are different pairwise combinations of elastic, confluent and divergent interactions. From a combinatorial viewpoint, we also explain that the asymptotic solitons of a [(M,N),(M?1,N+1)]-soliton solution are identified by a pair of Grassmannian permutations.     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Multi-soliton solutions for the coupled nonlinear Schr?dinger-type equations

Nonlinear Schr?dinger-type equations can model the nonlinear waves in fluids, plasmas, nonlinear optics and atmosphere. In this paper, integrable coupled nonlinear Schr?dinger-type equations are investigated. With the aid of symbolic computation, the equations are transformed into their bilinear forms, by virtue of which the multi-soliton solutions are derived. Soliton interactions are analyzed, the elastic interactions are seen, while the dark, anti-dark, M- and W-shape solitons are exhibited with some parameters selected. The propagating solitons can preserve their properties after the interaction, and the profiles of them depend on the corresponding dispersion relations. The amplitudes, velocities of the solitons are found to be influenced by the coefficient of the original equations, which is detailed in the paper.     

New double-periodic solutions of fractional Drinfeld–Sokolov–Wilson equation in shallow water waves

Nonlinear Dynamics - In the present article, the new exact solutions of time-fractional coupled Drinfeld–Sokolov–Wilson equations have been derived by using a new reliable analytical...     

Solitonic interactions,Darboux transformation and double Wronskian solutions for a variable-coefficient derivative nonlinear Schrödinger equation in the inhomogeneous plasmas

By symbolic computation we study a variable-coefficient derivative nonlinear Schrödinger (vc-DNLS) equation describing nonlinear Alfvén waves in inhomogeneous plasmas. Based on the Lax pair of the vc-DNLS equation, the N-fold Darboux transformation is constructed via a gauge transformation and the reduction technique. Multi-solitonic solutions in terms of the double Wronskian for the vc-DNLS equation are obtained. Two- and three-solitonic interactions are analyzed graphically, i.e., overtaking, head-on and parallel interactions. Plasma streaming and inhomogeneous magnetic field control the amplitudes and velocities of the solitonic waves, respectively. The nonuniform density affects the amplitudes of the solitonic waves. The effects of the spectral parameters on the dynamics of the two-solitonic waves are discussed. Our results might facilitate the analytic investigation on certain inhomogeneous systems in the Earth’s magnetosphere, solar winds, planetary bow shocks, dusty cometary tails and interplanetary shocks.     

Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup equations

Nonlinear Dynamics - In this paper, the Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup (WBK) equations are studied, which can help us increase the...     

An adaptive artificial viscosity for the displacement shallow water wave equation

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom,the moving wet-dry interface, and so on. In this paper, an adaptive artificial viscosity(AAV) is proposed and combined with the displacement shallow water wave equation(DSWWE) to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface. The effec...     

Functional and generalized separable solutions to unsteady Navier–Stokes equations

The paper studies unsteady Navier–Stokes equations with two space variables. It shows that the non-linear fourth-order equation for the stream function with three independent variables admits functional separable solutions described by a system of three partial differential equations with two independent variables. The system is found to have a number of exact solutions, which generate new classes of exact solutions to the Navier–Stokes equations. All these solutions involve two or more arbitrary functions of a single argument as well as a few free parameters. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary; such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for solving certain model problems and testing numerical and approximate analytical hydrodynamic methods. The paper uses the obtained results to describe some model unsteady flows of viscous incompressible fluids, including flows through a strip with permeable walls, flows through a strip with extrusion at the boundaries, flows onto a shrinking plane, and others. Some blow-up modes, which correspond to singular solutions, are discussed.     

Un théorème d'existence de solutions d'un problème de shallow water

Résumé Nous présentons un théorème d'existence de solutions d'un problème de type shallow water, en formulation hauteur-vitesse (h, u). Nous obtenons une majoration de type énergie et construisons des solutions approchées respectant cette majoration. La difficulté essentielle du passage à la limite provient de l'unique majoration connue de la hauteur h qui est du type: h et hlogh bornés dans L (0T;L ¹()). Mémoire presenté par P.-L. Lions     

Bright–dark vector soliton solutions for the coupled complex short pulse equations in nonlinear optics

Under investigation in this paper are the coupled complex short pulse equations, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the Hirota method, bright–dark one- and two-soliton solutions are obtained. Interactions between two bright or two dark solitons are verified to be elastic through the asymptotic analysis. With different parameter conditions, the oblique interactions, bound states of solitons and parallel solitons are analyzed.     

Weak in Space, Log in Time Improvement of the Lady?enskaja�CProdi�CSerrin Criteria

In this article we present a Ladyženskaja–Prodi–Serrin Criteria for regularity of solutions for the Navier–Stokes equation in three dimensions which incorporates weak L ^p norms in the space variables and log improvement in the time variable.     

Prediction of the boiling temperature and heat flux in sugar–water solutions under pool-boiling conditions

Condensing of a sugar–water solution is a widely used production process, especially in food industry. In this study, boiling temperature and heat transfer of different concentration levels of sugar/water solution is experimentally studied. In the experiment, the pool boiling with constant temperature difference between surface and boiling temperature is investigated. Boiling point of sugar/water solution depends on sugar mass concentration and on vapor phase pressure. A function is suggested to calculation the boiling temperature. The experimental data and the calculated values of boiling temperature are compared. The results are verified with previous investigations. It is determined that the heat flux between surface and sugar/water solution while pool boiling displays a linear relation with water mass concentration in the solution. Heat transfer coefficient could be determined in dependency of surface temperature and sugar mass concentration. Furthermore a function is suggested to predict the heat flux for engineering purpose, which is already used in similar form for pure substances.     

Properties of the Eulerian–Lagrangian method using linear interpolators in a three-dimensional shallow water model using z-level coordinates

We investigate the impact of the inexact interpolation on the Eulerian–Lagrangian solution of the advection equation by combining numerical experiments and formal analysis. The simulations, respectively, using the Eulerian–Lagrangian method (ELM) and the upwind scheme are compared. The artificial resistance of the ELM is observed which is characterised by the higher free-surface elevation and the distorted turbulent properties at a smaller time step. Through analysis, we find that the abnormalities are caused by the fact the conventional linear interpolation does not adapt well to the nonlinear velocity distribution, which produces an advection computation error that increases with a decreasing time step. The phenomena are explained and an improved method ELM is proposed based on the illustrations and analysis. The new method combines the face-controlled interpolation and the adjustable sub time steps to skip the large computation error domain in the backtracking, and it is validated by the original test case.     

Wronskian solutions and integrability for a generalized variable-coefficient forced Korteweg–de Vries equation in fluids

Under investigation in this paper is a generalized variable-coefficient forced Korteweg–de Vries equation, which can describe the shallow-water waves, internal gravity waves, and so on. With symbolic computation, the soliton solutions in the Wronskian form are derived based on the given bilinear form. Bäcklund transformation and Lax pair for such equation are also constructed. Variable coefficients and parameters of three solitons are managed to observe the features of the solitonic propagation and interaction, e.g., the solitonic velocity, amplitude and background. Our results could be expected to benefit the relevant problems in fluids.     

Fourier transform profilometry for water waves: how to achieve clean water attenuation with diffusive reflection at the water surface?

We present a study of the damping of capillary-gravity waves in water containing pigments. The practical interest comes from a recent profilometry technique (FTP for Fourier Transform Profilometry) using fringe projection onto the liquid-free surface. This experimental technique requires diffusive reflection of light on the liquid surface, which is usually achieved by adding white pigments. It is shown that the use of most paint pigments causes a large enhancement of the damping of the waves. Indeed, these paints contain surfactants which are easily adsorbed at the air–water interface. The resulting surface film changes the attenuation properties because of the resonance-type damping between capillary-gravity waves and Marangoni waves. We study the physicochemical properties of coloring pigments, showing that particles of the anatase (TiO₂) pigment make the water surface light diffusive while avoiding any surface film effects. The use of the chosen particles allows to perform space-time resolved FTP measurements on capillary-gravity waves, in a liquid with the damping properties of pure water.     

“Simple” solutions of the equations of dynamics for a polytropic gas

The notion of a “simple” solution of a system of differential equations that admit a local Lie group G of transformations of the basic space is considered as an invariant H-solution of type (0, 0) with respect to the subgroup HυG. Such solutions are attractive since they are described by explicit formulas that provide a clear physical interpretation for them. For gas-dynamic equations with a polytropic gas law, all simple solutions that are not related to special forms of gas flow are listed. Examples of simple solutions are given and the collapse phenomenon, which has been previously studied for barochronic flows, is described. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 5–12, March–April, 1999.     

On the Navier�CStokes equations with rotating effect and prescribed outflow velocity

We consider the equations of Navier–Stokes modeling viscous fluid flow past a moving or rotating obstacle in \mathbb R^d{\mathbb R^d} subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use L ^p -techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in \mathbb R^d{\mathbb R^d} is solved by an evolution system on L^p_s(\mathbb R^d){L^p_{\sigma}(\mathbb R^d)} for 1 < p < ∞. For this we use results about time-dependent Ornstein–Uhlenbeck operators. Finally, we prove, for p ≥ d and initial data u₀ ? L^p_s(\mathbb R^d){u_0\in L^p_{\sigma}(\mathbb R^d)}, the existence of a unique mild solution to the full Navier–Stokes system.     

Does generalized elastica lead to bimodal optimal solutions?

This paper answers the following question. A compressed rod clamped at both ends is assumed to rotate with a constant angular velocity. In the sense of classical Bernoulli–Euler elastica theory, the shape of the lightest rod, stable against buckling, is bimodal (i.e. associated with two buckling modes). What will be the case if we introduce more physical information in the rod model by assuming that it can suffer not only flexure but also compression and shear?     

Solitary wave solutions of the nonlinear generalized Pochhammer–Chree and regularized long wave equations

In this paper, we implement some fast and high accuracy numerical algorithms to obtain the solitary wave solutions of generalized Pochhammer?CChree (PC) and regularized long wave (RLW) equations. We employ the discrete Fourier transform to discretize the original partial differential equations (PDEs) in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. The proposed methods are fast and accurate due to the use of the fast Fourier transform in combination with explicit fourth-order time stepping methods. For RLW equation we investigate the propagation of a single solitary and interaction of two and three solitary waves. Moreover, three invariants of motion (mass, energy, and momentum) are evaluated to determine the conservation properties of the problem, and the numerical schemes lead to accurate results. The numerical results are compared with analytical solutions and with those of other recently published methods to confirm the accuracy and efficiency of the presented schemes.