Two-dimensional non-linear evolution equations: The derivation and the transient wave solution

The general theory of two-dimensional evolution equations describing transient wave propagation in non-linear continuous media is presented. The ray method is used and the two-dimensional evolution equations for plane and cylindrical wave-beams are obtained. The transient wave solutions are discussed briefly. A transformation of variables is proposed that permits the transformation of the two-dimensional evolution equation into a one-dimensional evolution equation with coordinate-dependent coefficients. A breakdown time analysis is carried out for this case. The dispersion relations for plane and cylindrical wave-beams are presented. The non-linear dispersion relation is obtained by making use of a series representation.     

On the derivation of some non-linear evolution equations and their progressive wave solutions

In the present work, utilizing the reductive perturbation method, the non-linear equations of a prestressed viscoelastic thick tube filled with a viscous fluid are examined in the longwave approximation and some evolution equations and their modified forms are derived. The analytical solution of some of these equations are obtained and it is shown that for perturbed cases, the wave amplitude and the phase velocity decay in the time parameter.     

Multiple scale perturbation for second-order non-linear differential equations

Asymptotic solutions of a class of second-order non-linear differential equations with variable coefficients are studied. For large values of the parameter, the differential equations are of the singular-perturbation type and approximations are constructed by the generalized method of multiple scales.     

Modified dissipativity for a non-linear evolution equation arising in turbulence

We are concerned with the regularity properties for all times of the equation $$\frac{{\partial U}}{{\partial t}}\left( {t,x} \right) = - \frac{{\partial ^2 }}{{\partial x^2 }}\left[ {U\left( {t,{\text{0}}} \right) - U\left( {t,x} \right)} \right]^2 - v\left( { - \frac{{\partial ^2 }}{{\partial x^2 }}} \right)^\alpha U\left( {t,x} \right)$$ which arises, with α=1, in the theory of turbulence. Here U(t,·) is of positive type and the dissipativity α is a non-negative real number. It is shown that for arbitrary ν≧0 and ?>0, there exists a global solution in \(L^\infty [0,\infty ;H^{\tfrac{3}{2} - \varepsilon } (\mathbb{R})]\) . If ν>0 and \(\alpha > \alpha _{cr} = \tfrac{1}{2}\) , smoothness of initial data persists indefinitely. If 0≦α<α _cr, there exist positive constants ν₁(α) and ν₂(α), depending on the data, such that global regularity persists for ν>ν₁(α), whereas, for 0≦ν<ν₂(α), the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of α _cr, similar results hold for the Navier-Stokes equation.     

Linear and non-linear iterative methods for the incompressible Navier-Stokes equations

In this study, the discretized finite volume form of the two-dimensional, incompressible Navier-Stokes equations is solved using both a frozen coefficient and a full Newton non-linear iteration. The optimal method is a combination of these two techniques. The linearized equations are solved using a conjugate-gradient-like method (CGSTAB). Various types of preconditioning are developed. Completely general sparse matrix methods are used. Investigations are carried out to determine the effect of finite volume cell anisotropy on the preconditioner. Numerical results are given for several test problems.     

A new multigrid algorithm for non-linear equations in conjunction with time-marching procedures

Full approximate storage (FAS) multigrid algorithm is the most commonly used multigrid algorithm for non-linear equations. The algorithm initially developed for steady-state equations was later extended to obtain steady-state solutions employing unsteady equations. In extending the FAS algorithm for the steady-state non-linear equations to unsteady non-linear equations, the FAS algorithm does not to take into account that the governing equations are typically linearized in time before they are solved. Thus, there is a scope to develop a new multigrid algorithm to apply the multigrid technique to the equations linearized in time. In the present work, such an algorithm is developed exploring this possibility and is implemented for two-dimensional incompressible and compressible flows coupled with explicit time marching procedures. The results of the new algorithm compare favourably with those of the FAS multigrid method and single grid.     

Exact solutions in 1+1 dimensions of the general two-velocity discrete illner model

The Illner model is the most general two-velocity model of the discrete Boltzmann equation. It includes, as particular cases, both the Carleman and the McKean model. Exact solutions in 1+1 dimensions of the general two-velocity discrete Illner model can be studied in a concise way. The conclusions of the precursors need ameliorating. A new type of exact solutions in 1+1 dimensions is obtained. This gives a general method for studying non-trivial exact solutions for the similar discrete Boltzmann equation. Project supported by the National Natural Science Foundation of China (19631060) and the China Post-Doctoral Science Foundation     

Homoclinic orbits for some (2+1)-dimensional nonlinear Schr(o)dinger-like equations

Chaos is closely associated with homoclinic orbits in deterministic nonlinear dynamics. In this paper, analytic expressions of homoclinic orbits for some (2 1)dimensional nonlinear Schr6dinger-like equations are constructed based on Hirota's bilinear method, including long wave-short wave resonance interaction equation, generalization of the Zakharov equation, Mel'nikov equation, and g-Schr(o)dinger equation are constructed based on Hirota's bilinear method.     

Analytical solutions for some nonlinear evolution equations

IntroductionItiswell_knownthatmanyimportantdynamicsprocessescanbedescribedbyspecificnonlinearpartialdifferentialequations .Whenanonlinearpartialdifferentialequationisusedtodescribeaphysicalparameterthatshowssomekindsofpropagationoraggregationproperties,oneofthemostimportantphysicalmotivationsistosolvethepartialdifferentialequationwithacertaintypeoftravellingwavesolution .Inthepastseveraldecades,therehavebeenmanyattemptsinthisfieldbothbymathematiciansandphysicists[1]- [16 ],however,duetothecomp…     

Taylor expansion method for nonlinear evolution equations

Introduction Thestudyofnonlinearevolutionequationsisafascinatingproblemwhichisattheveryheart oftheunderstandingofmanyimportantproblemsinthenaturalsciences[1,2].Thenonlinear evolutionequationsandtheirnumericalapproximationareveryimportantintheareasof theoreticalmathematicsandcomputationalmathematics.Aninterestingfeatureofthe approximationtheoryofthenonlinearevolutionequationsistheapplicationsofthefunctional analyticmethodstothenumericalapproximationofthenonlinearevolutionequations. Thispaperist…     

On the equivalence of simplest non-linear rheological equations for viscoelastic polymer media

Some equivalence conditions are formulated for non-linear models of polymer melts and solutions that are analogous to known conditions for three-constant linear rheological equations. The resulting model is analysed in simple shear and elongational flows. The kinematics of finite elastoviscous strains is considered in an appendix.     

Travelling fronts in non-local evolution equations

The existence of travelling fronts and their uniqueness modulo translations are proved in the context of a one-dimensional, non-local, evolution equation derived in [5] from Ising systems with Glauber dynamics and Kac potentials. The front describes the moving interface between the stable and the metastable phases and it is shown to attract all the profiles which at ± are in the domain of attraction of the stable and, respectively, the metastable states. The results are compared with those of Fife & McLeod [13] for the Allen-Cahn equation.     

Localized coherent structures of the (2+1)-dimensional higher order Broer-Kaup equations

ＩｎｔｒｏｄｕｃｔｉｏｎＳｏｌｉｔｏｎｉｓａｃｏｍｐｌｉｃａｔｅｄｍａｔｈｅｍａｔｉｃａｌｓｔｒｕｃｔｕｒｅｂａｓｅｄｏｎｔｈｅｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎ[1].Ｔｈｏｕｇｈｓｏｌｉｔｏｎｓｔｒｕｃｔｕｒｅｓａｎｄｐｒｏｐｅｒｔｉｅｓｏｆｔｈｅ ( 1 + 1 )＿ｄｉｍｅｎｓｉｏｎａｌｎｏｎｌｉｎｅａｒｐｈｙｓｉｃａｌｍｏｄｅｌｓｈａｖｅｂｅｅｎｓｔｕｄｉｅｄｗｅｌｌａｎｄｕｎｄｅｒｓｔｏｏｄｆｕｒｔｈｅｒ,ｔｈｅｓｏｌｉｔｏｎｓｔｒｕｃｔｕｒｅｓｉｎｈｉｇｈｅｒｓｐａｔｉａｌｄｉ…