Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation

Combining the symplectic variations theory,the homogeneous control equa- tion and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced.Firstly,based on the generalized Hamilton variation principle,the non-homogeneous Hamilton canonical equation for piezothermoelastic bod- ies was derived.Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered,and the non-homogeneous canoni- cal equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the inceusement of dimensions of the canonical equation.For the convenience of deriving Hamilton isoparametric element formulations with four nodes,one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle.The homogeneous equa- tion simplifies greatly the solution programs which are often performed to solve non- homogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.     

The integrable Boussinesq equation and it’s breather,lump and soliton solutions

The fourth-order nonlinear Boussinesq water wave equation, which explains the propagation of long waves in shallow water, is explored in this article. We used the Lie symmetry approach to analyze the Lie symmetries and vector fields. Then, by using similarity variables, we obtained the symmetry reductions and soliton wave solutions. In addition, the Kudryashov method and its modification are used to explore the bright and singular solitons while the Hirota bilinear method is effectively used to obtain a form of breather and lump wave solutions. The physical explanation of the extracted solutions was shown with the free choice of different parameters by depicting some 2-D, 3-D, and their corresponding contour plots.

     

Bifurcations and exact solutions of an asymptotic rotation-Camassa–Holm equation

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth’s rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained.

     

Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - In this work, a non-isospectral and variable-coefficient Kadomtsev–Petviashvili equation is considered using Hirota’s bilinear form and a direct assumption with...     

The Toda System and Clustering Interfaces in the Allen–Cahn equation

We consider the Allen–Cahn equation in a bounded, smooth domain Ω in , under zero Neumann boundary conditions, where is a small parameter. Let Γ₀ be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions for Γ₀, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are and which collapse onto Γ₀ as . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.     

Group classification and exact solutions of a generalized Emden–Fowler equation

The aim of this work is to perform a complete symmetry classification of a generalized Emden-Fowler equation. The various forms of this equation are extensively studied in the literature and they have applications in astrophysical and physiological phenomena. The classical approach of group classification and the procedure based upon the Lie algebras of low dimension are employed for classification. Exact solutions of the invariant equations are derived.     

Nonlinear Schrdinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrdinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.     

Painlevé analysis and invariant solutions of generalized fifth-order nonlinear integrable equation

In present work, new form of generalized fifth-order nonlinear integrable equation has been investigated by locating movable critical points with aid of Painlevé analysis and it has been found that this equation passes Painlevé test for \(\alpha =\beta \) which implies affirmation toward the complete integrability. Lie symmetry analysis is implemented to obtain the infinitesimals of the group of transformations of underlying equation, which has been further pre-owned to furnish reduced ordinary differential equations. These are then used to establish new abundant exact group-invariant solutions involving various arbitrary constants in a uniform manner.     

Lumps and rogue waves of generalized Nizhnik–Novikov–Veselov equation

In this paper, via generalized bilinear forms, we consider the (\(2+1\))-dimensional bilinear p-Sawada–Kotera (SK) equation. We derive analytical rational solutions in terms of positive quadratic functions. Through applying the dependent transformation, we present a class of lump solutions of the (\(2+1\))-dimensional SK equation. Those rationally decaying solutions in all space directions exhibit two kinds of characters, i.e., bright lump wave (one peak and two valleys) and bright–dark lump wave (one peak and one valley). In addition, we also obtain three families of bright–dark lump wave solutions to the nonlinear p-SK equation for \(p=3\).     

A rheological equation for anisotropic–anelastic media and simulation of field seismograms

In many cases, geological formations are composed of layers of dissimilar properties whose thicknesses are small compared to the wavelength of the seismic signal, as for instance, a sandstone formation that has intra-reservoir thin mudstone layers. A proper model is represented by an anisotropic (transversely isotropic) and viscoelastic stress–strain relation. In this work, we consider a sandstone reservoir, such as the Utsira formation, saturated with CO₂ and use White’s mesoscopic model to describe the energy loss of the seismic waves. The mudstone layers are assumed to be isotropic, poroelastic and lossless. Then, Backus averaging provides the complex and frequency-dependent stiffnesses of the transversely isotropic (TI) long-wavelength equivalent medium. We obtain the associated wave velocities and quality factors as a function of frequency and propagation direction, while the synthetic seismograms are computed with a finite-element (FE) method in the space-frequency domain. In this way, the frequency-dependent properties of the medium are modeled exactly, without the need of approximations with viscoelastic mechanical models. Numerical simulations of synthetic seismograms show results in agreement with the predictions of the theories and significant differences due to attenuation and anisotropic effects compared to the ideal isotropic and lossless rheology.     

An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.     

Variational principles and conservation laws to the Burridge–Knopoff equation

We generate conservation laws for the Burridge–Knopoff equation which model nonlinear dynamics of earthquake faults by a new conservation theorem proposed recently by Ibragimov. One can employ this new general theorem for every differential equation (or systems) and derive new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to the Burridge–Knopoff equation.     

Bilinear form and soliton interactions for the modified Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics

In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transformation and auxiliary function, new bilinear form for the mKP equation is constructed, which is different from those in previous literatures. Based on the bilinear form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.     

Representations of ℂ, biharmonic vector fields,and the equilibrium equation of planar elasticity

The equilibrium equation for an elastic body subjected to surface forces asserts the linear dependence of the Laplacian and the gradient of the divergence of the vector field which gives the displacement at each point. James Clerk Maxwell (1831–1879) was the first to point out that the component functions of such a field are biharmonic, i.e., their Laplacians are harmonic functions. Using only algebraic tools familiar to advanced undergraduates we show that the usual complex variable representation of two-variable biharmonic functions falls naturally out of a power series construction based on matrix representations of . Under the assumption of linear stress and strain components, this construction is then used to describe the solutions to the planar equilibrium equation in terms of the geometry of the Moebius plane.     

Solution of flat crack problem by using variational principle and differential–integral equation

Variational principle is used to solve some flat crack problems in three-dimensional elasticity. In the formulation, the strain energy is evaluated by multiplying the crack opening displacement (COD) by the boundary traction. The boundary traction is related to the COD function by a differential–integral representation. By using an integration by part, the portion of the strain energy of the potential functional can be expressed by a repeated integral. In the integral all the integrated functions are non-singular. Letting the functional be minimum, the solution is obtained. In the actual solution, the COD function is represented by a shape function family in which several undetermined coefficients are involved. Using the variational principle, the coefficients are obtained. Several numerical examples are given with the stress intensity factors calculated along the crack border.     

A new type of multiple-lump and interaction solution of the Kadomtsev–Petviashvili I equation

Nonlinear Dynamics - In this paper, the solution in the form of Grammian of the Kadomtsev–Petviashvili I equation is employed to investigate a new type of multiple-lump solution. The bound...     

The modified Casson＇s equation and its application to pipe flows of shear-thickening fluid

A few additional data from our previous experiments were plotted to emphasize the shear-thickening behavior of deoxy sickle erythrocyte (SS) suspension. A constitutive equation (named as FX equation) was developed and applied to a cylindrical pipe flow of a shear-thickening fluid. A blunt velocity profile and its volume flow rate were calculated. The flow was non-viscous (potential) in the central part of the pipe (i.e. the central core or the central plug-flow), and became more and more viscous towards the wall of the pipe after a specific radial distance, which was determined by a critical shear rate of (named as Fungs shear rate). Furthermore, combining the FX equation with the original Cassons equation, the author obtained a modified Cassons equation by introducing .The English text was polished by Yunming Chen.     

Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation

Nonlinear Dynamics - In this paper, we use the bifurcation method of dynamical systems to investigate the nonlinear wave solutions of the modified Benjamin–Bona–Mahony equation. These...     

Riemann–Hilbert approach and N-soliton solutions for a new two-component Sasa–Satsuma equation

Nonlinear Dynamics - A new two-component Sasa–Satsuma equation associated with a $$4\times 4$$ matrix spectral problem is proposed by resorting to the zero-curvature equation....