Third-order Nonlinear Evolution Equation Research Articles

Thin Film Evaporation Modeling of the Liquid Microlayer Region in a Dewetting Water Bubble

Understanding the mechanism of bubble growth is crucial to modeling boiling heat transfer and enabling the development of technological applications, such as energy systems and thermal management processes, which rely on boiling to achieve the high heat fluxes required for their operation. This paper presents analyses of the evaporation of “microlayers”, i.e., ultra-thin layers of liquid present beneath steam bubbles growing at the heated surface in the atmospheric pressure nucleate of boiling water. Evaporation of the microlayer is believed to be a major contributor to the phase change heat transfer, but its evolution, spatio-temporal stability, and impact on macroscale bubble dynamics are still poorly understood. Mass, momentum, and energy transfer in the microlayer are modeled with a lubrication theory approach that accounts for capillary and intermolecular forces and interfacial mass transfer. The model is embodied in a third-order nonlinear film evolution equation, which is solved numerically. Variable wall-temperature boundary conditions are applied at the solid–liquid interface to account for conjugate heat transfer due to evaporative heat loss at the liquid–vapor interface. Predictions obtained with the current approach compare favorably with experimental measurements of microlayer evaporation. By comparing film profiles at a sequence of times into the ebullition cycle of a single bubble, likely values of evaporative heat transfer coefficients were inferred and found to fall within the range of previously reported estimates. The result suggests that the coefficients may not be a constant, as previously assumed, but instead something that varies with time during the ebullition cycle.

A deep learning method for solving third-order nonlinear evolution equations

It has still been difficult to solve nonlinear evolution equations analytically. In this paper, we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly. Specifically, the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters. In particular, numerical experiments on several third-order nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation, modified KdV equation, KdV–Burgers equation and Sharma–Tasso–Olver equation, demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.

Exact Traveling Wave Solutions of Nonlinear Evolution Equations: Indeterminant Homogeneous Balance and Linearizability

Exact traveling (solitary) wave solutions of nonlinear partial differential equations (NLPDEs) are analyzed for third-order nonlinear evolution equations. These equations have indeterminant homogenous balance and therefore cannot be solved by the Power Index Method (PIM). Some evolution equations are linearizable where solutions are transferred from those of a linear PDE. For other evolution equations transforming to a NLPDE which has a homogenous balance gives rise to possible solutions by the PIM. The solutions for evolution equations that are not linearizable are developed here.

Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation

In this work, the lump solution and the kink solitary wave solution from the (2 + 1) -dimensional third-order evolution equation, using the Hirota bilinear method are obtained through symbolic computation with Maple. We have assumed that the lump solution is centered at the origin, when t = 0 . By considering a mixing positive quadratic function with exponential function, as well as a mixing positive quadratic function with hyperbolic cosine function, interaction solutions like lump-exponential and lump-hyperbolic cosine are presented. A completely non-elastic interaction between a lump and kink soliton is observed, showing that a lump solution can be swallowed by a kink soliton.

A class of third-order nonlinear evolution equations admitting invariant subspaces and associated reductions

With the aid of symbolic computation by Maple, a class of third-order nonlinear evolution equations admitting invariant subspaces generated by solutions of linear ordinary differential equations of order less than seven is analyzed. The presented equations are either solved exactly or reduced to finite-dimensional dynamical systems. A number of concrete examples admitting invariant subspaces generated by power, trigonometric and exponential functions are computed to illustrate the resulting theory.

Galilei symmetries of KdV-type nonlinear evolution equations

We perform Galilei symmetry group classification of a class of third-order nonlinear evolution equations in one spatial variable, which generalize KdV and mKdV equations. All inequivalent PDEs belonging to the class in question which admit the classical Galilei group, the extended Galilei group and the natural extensions of the extended Galilei groups are constructed. The list of so obtained invariant equations may be used as motion equations for they all satisfy the Galilei relativity principle. In addition, we also give a complete classification of group invariant solutions for one obtained Galilei-invariant equation.

Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras

We give a complete point-symmetry classification of all third-order evolution equations of the form u t =F(t,x,u,u x ,u xx )u xxx +G(t,x,u,u x ,u xx ) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.

Smooth soliton solutions of a new integrable equation by Qiao

We find a transformation which relates a new third-order integrable nonlinear evolution equation, introduced recently by Qiao, with the well-known modified Korteweg–de Vries equation. Then we use this transformation to derive smooth soliton solutions of the new equation from the known rational and soliton solutions of the old one.

Waves, conservation laws and symmetries of a third-order nonlinear evolution equation

In this paper a less studied nonlinear partial differential equation of the third-order is under consideration. Important properties concerning advanced character such like conservation laws and the equation of continuity are given. Characteristic wave properties such like dispersion relations and both the group and phase velocities are derived explicitly. In addition, we discuss the non-classical case relating to potential symmetries for the first time. Further, for practical applications in several domains of sciences we discuss in detail approximate symmetries. Finally, as a further new contribution we deduce new generalized symmetries of lower order. Some important notes relating to future intensions are given. Key Words: Nonlinear partial differential equations, evolution equations, symmetries.

Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six nonequivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.

Integrability of a nonlinear evolution equation derived from isoperimetric plane curve motion

We provide a geometrical interpretation for the series of transformations used by Sakovich to map the third-order nonlinear evolution equation obtained by Chou and Qu to the mKdV equation. We also discuss its bi-Hamiltonian integrability as well as integrable equations associated with this system.

New exact solutions with an arbitrary function of two -dimensional nonlinear evolution equations

In this Letter, the multi-linear variable separation approach which is based on corresponding Bäcklund transformation (MLVSA-BT) is extended to solve a ( 1 + 1 ) -dimensional third-order nonlinear evolution equation and a ( 1 + 1 ) -dimensional differential-difference Toda-like equation. New exact solution with an arbitrary function is derived. Abundant coherent structures such as soliton and solitoff can be found by selecting corresponding lower-dimensional function appropriately.

Symmetry classification of KdV-type nonlinear evolution equations

Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semi-direct sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirty-eight, fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear third-order evolution equation.

On integrability of one third-order nonlinear evolution equation

We study one third-order nonlinear evolution equation, recently introduced by Chou and Qu in a problem of plane curve motions, and find its transformation to the modified Korteweg–de Vries equation, its zero-curvature representation with an essential parameter, and its second-order recursion operator.

Fourth-order nonlinear evolution equations for counterpropagating capillary-gravity wave packets on the surface of water of infinite depth

Asymptotically exact nonlocal fourth-order nonlinear evolution equations are derived for two counterpropagating capillary-gravity wave packets on the surface of water of infinite depth. On the basis of these equations a stability analysis is made for a uniform standing capillary-gravity wave for longitudinal perturbation. The instability conditions and an expression for the maximum growth rate of instability are obtained. Significant deviations are noticed between the results obtained from third-order and fourth-order nonlinear evolution equations.

Symmetry groups and spiral wave solution of a wave propagation equation

We study a third-order nonlinear evolution equation, which can be transformed to the modified KdV equation, using the Lie symmetry method. The Lie point symmetries and the one-dimensional optimal system of the symmetry algebras are determined. Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modified KdV equation. The group-invariant solutions, particularly the travelling wave and spiral wave solutions, are discussed in detail and a type of spiral wave solution which is smooth in the origin is obtained.

On the existence of a global strong radial symmetric solution for a third-order nonlinear evolution equation in two space dimensions

In this paper we consider the following nonlinear evolution equation: φ tt−Δφ− div 1 3 |∇φ| 2∇φ =Δφ t, x∈ R 2, t⩾0, with initial data φ( x,0)= φ 0( x), φ t ( x,0)= φ 1( x), φ 0∈H 3( R 2) , φ 1∈H 2( R 2) , which is an extension of a particular case of the unidimensional viscoelasticity equations. We prove the existence of a unique global strong solution φ∈C [0,+∞[;H 3 ∩C 1 [0,+∞[;H 2 ∩C 2 [0,+∞[;L 2 , without restrictions on the “size” of φ 0 and φ 1; if φ 0 and φ 1 only depend on r=| x|=( x 1 2+ x 2 2) 1/2, the solution φ only depends on r and t.

Long-time asymptotics of solutions of the third-order nonlinear evolution equation governing wave propagation in relaxing media

A classical Cauchy problem for a third-order nonlinear evolution equation is considered. This equation describes the propagation of weakly nonlinear waves in relaxing media. The global existence and uniqueness of its solutions is proved and the solution is constructed in the form of a series in a small parameter present in the initial conditions. Its long-time asymptotics is calculated, which shows the presence of two solitary wave pulses traveling in opposite directions and diffusing in space. Each of them is governed by Burgers’ equation with a transfer.