Nonlinear Third-order Partial Differential Equations Research Articles

$ H^1 $ solutions for a modified Korteweg-de Vries-Burgers type equation

<p>This paper modeled the dynamics of microbubbles coated with viscoelastic shells using the modified Korteweg-de Vries-Burgers equation, a nonlinear third-order partial differential equation. This study focused on the well-posedness of the Cauchy problem associated with this equation.</p>

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

On the New Wave Behaviors of the Gilson-Pickering Equation

In this article, we study the fully non-linear third-order partial differential equation, namely the Gilson–Pickering equation. The $\left(1/{{G}^{'}}\right)$-expansion method, and the generalized exponential rational function method are used to construct various exact solitary wave solutions for a given equation. These methods are based on a homogeneous balance technique that provides an order for the estimation of a polynomial-type solution. In order to convert the governing equation into a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. As a result, we have constructed a variety of solitary wave solutions, such as singular solutions, compound singular solutions, complex solutions, topological, and non-topological solutions. Besides, the 2D, 3D and contour surfaces are plotted for all obtained solutions by choosing appropriate parameter values.

On a nonlinear third-order equation

A nonclassical nonlinear third-order partial differential equation modeling nonstationary processes in a semiconductor medium is studied. Seven classes of exact solutions of this equation given as explicit, implicit, and quadrature formulas are constructed. It is shown that, among these solutions, there are some bounded globally with respect to time and some approaching infinity on finite time intervals. These results are of interest in view of the following property of initial boundary-value problems for the given equation. Here the method of energy estimates used in many papers to justify the boundedness or unboundedness of solutions does not provide sufficient conditions for the unboundedness of solutions on finite time intervals.

SOLITARY TRAVELING WATER WAVES OF MODERATE AMPLITUDE

We prove the existence of solitary traveling wave solutions for an equation describing the evolution of the free surface for waves of moderate amplitude in the shallow water regime. This nonlinear third-order partial differential equation arises as an approximation of the Euler equations, modeling the unidirectional propagation of surface water waves. We give a description of the solitary wave profiles by performing a phase plane analysis and study some qualitative features of the solutions.

Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane

We prove existence and uniqueness results for nonlinear third-order partial differential equations of the form

The Spectrum of Coupled Random Matrices

The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy. In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction. Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations. These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices. Comment: 56 pages, published version, abstract added in migration

Symmetries of a class of nonlinear third-order partial differential equations

In this paper, we study symmetry reductions of a class of nonlinear third-order partial differential equations u t − ϵu xxt + 2 κu x = uu xxx + αuu x + βu x u xx , (1) where ϵ, κ, α, and β are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case, the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters ϵ = 1, α = −1, β = 3, and κ = 1 2 , admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters ϵ = 0, α = 1, β = 3, and κ = 0, admits a “compacton” solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters ϵ = 1, α = − 3, and β = 2, has a “peakon” solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.

Factorization and Painleve analysis of a class of nonlinear third-order partial differential equations

A class of fully nonlinear third-order partial differential equations (PDES) is considered. This class contains several examples which have recently appeared in the literature and for which rather unusual travelling-wave solutions have been given. These solutions consist essentially of sums of exponentials; we trace the occurrence of these exponentials back to the existence of a linear subequation which appears as a factor in the travelling-wave reduction. In addition, we consider the Painleve analysis of this set of equations, both for the original PDE and also for reductions to ordinary differential equations (ODES). No equation in the class considered survives the combination of PDE and ODE tests. Also, an equation in the class considered which is known to be integrable is shown to possess only the weak Painleve property. Our analysis, therefore, confirms the limitations of the Painleve test as a test for complete integrability when applied to fully nonlinear PDES.

A Theory of Post-Stall Transients in Axial Compression Systems: Part I—Development of Equations

An approximate theory is presented for post-stall transients in multistage axial compression systems. The theory leads to a set of three simultaneous nonlinear third-order partial differential equations for pressure rise, and average and disturbed values of flow coefficient, as functions of time and angle around the compressor. By a Galerkin procedure, angular dependence is averaged, and the equations become first order in time. These final equations are capable of describing the growth and possible decay of a rotating-stall cell during a compressor mass-flow transient. It is shown how rotating-stall-like and surgelike motions are coupled through these equations, and also how the instantaneous compressor pumping characteristic changes during the transient stall process.