Painlevé analysis of nonlinear evolution equations—an algorithmic method

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

We give a review of our recent results on group classification of the most general nonlinear evolution equation in one spatial variable. The method applied relies heavily on the results of our paper Acta Appl. Math., 69, 2001, in which we obtain the complete solution of group classification problem for general quasilinear evolution equation.

A fast compact time integrator method for a family of general order semilinear evolution equations

We investigate the integrability properties of discrete systems and singular integral evolution equations, showing that they correspond to different reductions and different concrete realisations of the same abstract algebraic structures. In particular, we derive the bihamiltonian formulation of prototypes examples like the discrete sine-Gordon equation and the sine-Hilbert equation.KeywordsConcrete RealisationTrace FormPauli MatrixHeiSenberg SpinPoisson TensorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Evolution equations and size distributions in nanocrystalline grain growth

In this paper, we consider the evolution equation 𝑢𝑡 = [𝑚 + 𝑘𝜓(𝑢)]𝑢, 𝑡 > 0 where 𝑡 represents dimensionless time and 𝜓(𝑢) is given function of 𝑢. We focus attention on the case when 𝜓(𝑢) = 𝑘(𝑢 − 𝑎)(𝑢 − 1), while 𝑚 and 𝑘 are constants.

The most general second order evolution equation ψt+F(x,t,ψψ*,ψx,ψx*,ψxx, ψxx*)=0, invariant under the Galilei, Galilei-similitude, and Schrödinger groups in two dimensions, is constructed. A preliminary step is a classification of all possible realizations of the corresponding Lie algebras of vector fields in ℛ2×𝒞 parametrized by x, t, ψ, and ψ*. Applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories. Among the Schrödinger invariant equations, in particular, are found integrable equations, linearizable by contact transformations.

We analyze the general nonlinear evolution equations for multigluon correlators derived in a previous paper by restricting ourselves to a double logarithmic region. In this region our evolution equation becomes local in transverse momentum space and amenable to simple analysis. It provides a complete nonlinear generalization of the Gribov-Levin-Ryskin equation. We find that the full double log evolution at high density becomes strikingly different from its linear doubly logarithmic Dokshitzer-Gribov-Lipatov-Alterelli-Parisi counterpart. An effective mass is induced by the nonlinear corrections which at high densities slows down the evolution considerably. We show that at small values of impact parameter the gluonic density grows as a logarithm of energy. At higher values of impact parameter the growth is faster, since the density of gluons is lower and nonlinearities are less important.

Averaging principle and hyperbolic evolution equations

Analytic study of the fifth order integrable nonlinear evolution equations by using the tanh method

The authors give a natural combination of partial-bar and Riemann-Hilbert problem inverse methods for an n th-order scalar spectral problem which solves a number of integrable non-linear evolution equations in two space and one time (2+1) dimensions. The theory embraces the two Kadomtsev-Petviashvili equations (1970) and their lump solutions.

In the symmetry approach framework, we solve the problem of classifying third-order integrable vector evolution equations that have zeroth-order conserved densities. We obtain the complete list of nine equations of this form. Two equations in the list were previously unknown. We find auto-Backlund transformations for the new equations.

We employ basic non-equilibrium thermodynamics to propose a general equation for the mean grain size evolution in a deforming medium, under the assumption that the whole grain size distribution remains self-similar. We show that the grain size reduction is controlled by the rate of mechanical dissipation in agreement with recent findings. Our formalism is self consistent with mass and energy conservation laws and allows a mixed rheology. As an example, we consider the case where the grain size distribution is lognormal, as is often experimentally observed. This distribution can be used to compute both the kinetics of diffusion between grains and of dynamic recrystallization. The experimentally deduced kinetics of grain size coarsening indicates that large grains grow faster than what is assumed in classical normal grain growth theory. We discuss the implications of this model for a mineral that can be deformed under both dislocation creep and grain size sensitive diffusion creep using experimental data of olivine. Our predictions of the piezometric equilibrium in the dislocation-creep regime are in very good agreement with the observations for this major mantle-forming mineral. We show that grain size reduction occurs even when the average grain size is in diffusion creep, because the largest grains of the grain size distribution can still undergo recrystallization. The resulting rheology that we predict for olivine is time-dependent and more non-linear than in dislocation creep. As the deformation rate remains an increasing function of the deviatoric stress, this rheology is not localizing.

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).

We use a statistical approach to investigate the modulational instability (Benjamin-Feir instability) in several nonlinear discrete systems: the discrete nonlinear Schrodinger (NLS) equation, the Ablowitz-Ladik equation, and the discrete deformable NLS equation. We derive a kinetic equation for the two-point correlation function and use a Wigner-Moyal transformation to write it in a mixed space-wave-number representation. We perform a linear stability analysis of the resulting equation and discuss the obtained integral stability condition using several forms of the initial unperturbed spectrum (Lorentzian and δ-spectrum). We compare the results with the continuum limit (the NLS equation) and with previous results.