Nonlinear Second Order Evolution Equations Research Articles

On stability and convergence of a three-layer semi-discrete scheme for an abstract analogue of the Ball integro-differential equation

We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.

A Galerkin approach to solitary wave propagation for the second-order nonlinear evolution equation based on quartic B-spline functions

This article aims to produce efficient numerical results using the Galerkin approach and quartic B-spline functions for the one-dimensional second-order nonlinear evolution equation. The quartic B-spline functions are used as the shape and weight functions for the Galerkin method. The time derivative and space parameters are discretized by the finite difference and Crank–Nicolson schemes, respectively. A Galerkin approach is investigated by propagating waves at various times and comparisons are made with published papers. The obtained results show that the scheme presented good results and conservations laws are all adequately obeyed.

Solving second-order nonlinear evolution partial differential equations using deep learning**The project is supported by the National Natural Science Foundation of China (No. 11675054), Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers’ equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions.

Group resonant interactions between surface and internal gravity waves in a two-layer system

Nonlinear interactions between surface and internal gravity waves in a two-layer system are studied using explicit second-order nonlinear evolution equations in Hamiltonian form. Motivated by the detailed experiment of Lewis, Lake & Ko (J. Fluid Mech., vol. 63, 1974, pp. 773–800), our focus is on surface wave modulation by the group resonance mechanism that corresponds to near-resonant triad interactions between a long internal wave and short surface waves. Our numerical solutions show good agreement with laboratory measurements of the local wave amplitude and slope, and confirm that the surface modulation becomes significant when the group velocity of the surface waves matches the phase speed of the internal wave, as the linear modulation theory predicts. It is shown, however, that, after the envelope amplitude is increased sufficiently, the surface and internal waves start to exchange energy through near-resonant triad interactions, which is found to be crucial to accurately describe the long-term surface wave modulation by an internal wave. The reduced amplitude equations are also adopted to validate this observation. For oceanic applications, numerical solutions are presented for a density ratio close to one and it is found that significant energy exchanges occur through primary and successive resonant triad interactions.

Equivalence groupoids of classes of nonlinear second-order evolution equations

Equivalence groupoids of classes of nonlinear second-order evolution equations

A family of wave equations with some remarkable properties.

We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found for two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair, we construct a Miura-type transformation relating the original equation to the Korteweg-de Vries (KdV) equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schrödinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second-order nonlinear evolution equation.

Stability of solutions to nonlinear wave equations with switching time delay

In this paper we study well-posedness and asymptotic stability for a class of nonlinear second-order evolution equations with intermittent delay damping. More precisely, a delay feedback and an undelayed one act alternately in time. We show that, under suitable conditions on the feedback operators, asymptotic stability results are available. Concrete examples included in our setting are illustrated. We give also stability results for an abstract model with alternate positive-negative damping, without delay.

Group classification of nonlinear evolution equations: Semi-simple groups of contact transformations

We generalize and modify the group classification approach of Zhdanov and Lahno (1999) to make it applicable beyond Lie point symmetries. This approach enables obtaining exhaustive classification of second-order nonlinear evolution equations in one spatial dimension invariant under semi-simple groups of contact transformations. What is more, all inequivalent second-order nonlinear evolution equations which admit semi-simple groups or groups having nontrivial Levi decompositions are constructed in explicit forms.

On the Evolutionary Fractionalp-Laplacian

In this work, existence results on nonlinear first order as well as doubly nonlinear second-order evolution equations involving the fractional |$p$|-Laplacian are presented. The proofs do not exploit any monotonicity assumption but rely on a compactness argument in combination with regularity of the Galerkin scheme and the nonlocal character of the operator.

Full discretisation of second-order nonlinear evolution equations: strong convergence and applications

Abstract Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

A nonlinear second-order evolution equation – classical and nonclassical symmetry analyses and their physical implications

A nonlinear second-order evolution equation – classical and nonclassical symmetry analyses and their physical implications

Faedo-Galerkin method for nonlinear second-order evolution equations with Volterra operators

We consider a class of second-order operator differential equations with \(w_{\lambda _0 } \)-pseudomonotone operators. The problem of the existence of solutions of the Cauchy problem for the these equations is investigated by the Faedo-Galerkin method. We establish important a priori estimates and give an example that illustrates the result obtained.

Capillary-driven flows along rounded interior corners

The problem of low-gravity isothermal capillary flow along interior corners that are rounded is revisited analytically in this work. By careful selection of geometric length scales and through the introduction of a new geometric scaling parameter for perfectly wetting fluids. The scaling and analysis of the problem captures much of the intricate geometric dependence of the viscous resistance and significantly reduces the reliance on numerical data compared with several previous solution methods and the numerous subsequent studies that cite them. In general, three asymptotic regimes may be identified from the large second-order nonlinear evolution equation: (I) the ‘sharp-corner’ regime, (II) the narrow-corner ‘rectangular section’ regime, and (III) the ‘thin film’ regime. Flows are observed to undergo transition between regimes, or they may exist essentially in a single regime depending on the system. Perhaps surprisingly, for the case of imbibition in tubes or pores with rounded interior corners similarity solutions are possible to the full equation, which is readily solved numerically. Approximate analytical solutions are also possible under the constraints of the three regimes, which are clearly identified. The general analysis enables analytic solutions to many rounded-corner flows, and example solutions for steady flows, perturbed infinite columns, and imbibing flows over initially dry and prewetted surfaces are provided.

Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, II

New results concerning Lie symmetries of nonlinear reaction-diffusion-convection equations, which supplement in a natural way the results published in the European Journal of Applied Mathematics (9(1998) 527–542) are presented.

Nonlocal symmetry and generation of solutions for reaction-diffusion equations

A class of second-order nonlinear evolution equations is considered. A new equation of this class which can be linearized has been found. Invariant nonlocal transformations and nonlocal superposition principle have been constructed for this equation. Nonlocal invariant transformations for some equations of this class also were studied. There are shown examples of generation of partial solutions.

Attractors for Second-Order Evolution Equations with a Nonlinear Damping

We study long-time dynamics of abstract nonlinear second-order evolution equations with a nonlinear damping. Under suitable hypotheses we prove existence of a compact global attractor and finiteness of its fractal dimension. We also show that any solution is stabilized to an equilibrium and estimate the rate of the convergence which, in turn, depends on the behaviour at the origin of the function describing the dissipation. If the damping is bounded below by a linear function, this rate is exponential. Our approach is based on far reaching generalizations of the Ceron–Lopes theorem on asymptotic compactness and Ladyzhenskaya’s theorem on the dimension of invariant sets. An application of our results to nonlinear damped wave and plate equations allow us to obtain new results pertaining to structure and properties of global attractors for nonlinear waves and plates.

Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms

New Lie and conditional symmetries of nonlinear diffusion equations with convection terms are constructed. Examples of new ansätze and exact solutions for some nonlinear equations (in particular, for the Murray and the Newell–Whitehead equations) are presented.

Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear evolution equations

Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear evolution equations

An evolution equation modeling inversion of tulip flames

We attempt to reduce the number of physical ingredients needed to model the phenomenon of tulip-flame inversion to a bare minimum. This is achieved by synthesising the nonlinear, first-order Michelson-Sivashinsky (MS) equation with the second order linear dispersion relation of Landau and Darrieus, which adds only one extra term to the MS equation without changing any of its stationary behaviour and without changing its dynamics in the limit of small density change when the MS equation is asymptotically valid. However, as demonstrated by spectral numerical solutions, the resulting second-order nonlinear evolution equation is found to describe the inversion of tulip flames in good qualitative agreement with classical experiments on the phenomenon. This shows that the combined influences of front curvature, geometric nonlinearity and hydrodynamic instability (including its second-order, or inertial effects, which are an essential result of vorticity production at the flame front) are sufficient to reproduce the inversion process.

Two-wave propagation and interaction for weakly nonlinear waves in a dissipative medium

An asymptotic approach is presented to study the propagation of two wave modes in a dissipative medium before their separation, as predicted by the theory based on the Burgers equation. This approach involves introducing a second-order nonlinear evolution equation of a special kind, which can be considered as a second-order Burgers equation. Some analytical solutions for this equation are found and discussed. An application of the general theory to MHD waves in a medium of finite conductivity is presented.