Time-space integrable decompositions of nonlinear evolution equations

Separation of the time and space variables of evolution equations is analyzed, without using any structure associated with evolution equations. The resulting theory provides techniques for constructing time-space integrable decompositions of evolution equations, which transform an evolution equation into two compatible Liouville integrable ordinary differential equations in the time and space variables. The techniques are applied to the KdV, MKdV and diffusion equations, thereby yielding several new time-space integrable decompositions of these equations.     

Classical Symmetry Reductions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions

Classical reductions of a (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation are classified. These reductions to systems of partial differential equations in 1+1 dimensions admit symmetries that lead to further reductions, i.e., to systems of ordinary differential equations. All these systems have been reduced to second-order ordinary differential equations. We present some particular solutions involving two arbitrary functions.     

Frobenius integrable decompositions for high-order nonlinear evolution equations

Two classes of nonlinear evolution equations are obtained, which possess Frobenius integrable decompositions. The resulting solitons are illustrated to describe the solution phenomena.     

Modelling of stochastic heat transfer in a solid

The unsteady partial differential equations for expectation and correlation distributions of the stochastic temperature distribution in a solid are obtained, when the coefficients and the source term in the stochastic heat transfer equations are white Gaussian processes. Some solutions of the unsteady partial differential equations for expectation and correlation distributions of stochastic heat transfer are presented.     

The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.     

Computation of local symmetries of second-order ordinary differential equations by the Cartan equivalence method

The Cartan equivalence method is used to find out if a given equation has a nontrivial Lie group of point symmetries. In particular, we compute invariants that permit one to recognize equations with a three-dimensional symmetry group. An effective method to transform the Lie system (the system of partial differential equations to be satisfied by the infinitesimal point symmetries) into a formally integrable form is given. For equations with a three-dimensional symmetry group, the formally integrable form of the Lie system is found explicitly. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 75–91, July, 1996.     

基于Frobenius定理的Hamilton-Jacobi方法的几何解释

给出了一阶偏微分方程特征微分方程组的一种基于Frobenius定理的几何解释,通过研究发现根据Frobenius定理可以从一阶偏微分方程直接得到其特征微分方程组;在此基础上说明如何利用几何方法从Hamilton正则方程出发找到与之对应的Hamilton-Jacobi方程.这种方法可以被用于非保守或非完整Hamilton力学问题的研究中,经典Hamilton-Jacobi方法是这种方法的一个特例.     

Degenerate non-stationary differential equations with delay in Banach spaces

A generalization of the operator method by Grisvard is used to ensure weak and strict solutions to some degenerate differential equations with delay in Banach spaces, whose operator coefficients are time depending. Some applications to ordinary and partial differential equations with delay are described.     

Well-posedness of SDEs with drifts in mixed-norm spaces and driven by mixed-noises

We prove the strong well-posedness for stochastic differential equations driven by partial α-stable noise and partial Brownian noise. The drift coefficients belong to certain mixed-norm spaces, which generalize the known results even for stochastic differential equations with single noise cases.     

Integrable Quasilinear Equations

We develop a classification scheme for integrable third-order scalar evolution equations using the symmetry approach to integrability. We use this scheme to study quasilinear equations of a particular type and prove that several equations that were suspected to be integrable can be reduced to the well-known Korteweg–de Vries and Krichever–Novikov equations via a Miura-type differential substitution.     

ADJOINT SYMMETRY CONSTRAINTS OF MULTICOMPONENT AKNS EQUATIONS

A soliton hierarchy of multicomponent AKNS equations is generated from an arbitraryorder matrix spectral problem,along with its bi-Hamiltonian formulation.Adjoint symmetryconstraints are presented to manipulate binary nonlinearization for the associated arbitraryorder matrix spectral problem.The resulting spatial and temporal constrained fiows are shownto provide integrable decompositions of the multicomponent AKNS equations.     

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.     

Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3 × 3 stencil

Hirota's bilinear method (‘direct method’) has been very effective for constructing soliton solutions to many integrable equations. The construction of one-soliton solution (1SS) and two-soliton solution (2SS) is possible even for non-integrable bilinear equations, but the existence of a generic three-soliton solution (3SS) imposes severe constraints and is in fact equivalent to integrability. This property has been used before in searching for integrable partial differential equations, and in this paper we apply it to two-dimensional (2D) partial difference equations defined on a 3 × 3 stencil. We also discuss how the obtained equations are related to projections and limits of the 3D master equations of Hirota and Miwa, and find that sometimes a singular limit is needed.     

Charpit System of Partial Differential Equations with Algebraic Constraints

In this paper the Charpit system of partial differential equations with algebraic constraints is considered. So, first the compatibility conditions of a system of algebraic equations and also of the Charpit system of partial differential equations are separately considered. For the combined system of equations of both types sufficient conditions for the existence of a solution are found. They lead to an algorithm for reducing the combined system to a Charpit system of partial differential equations of dimension less than the initial system and without algebraic constraints. Moreover, it is proved that this system identically satisfies the compatibility conditions if so does the initial system.     

An L 2-theory for a class of SPDEs driven by Lévy processes

In this paper we present an L 2-theory for a class of stochastic partial differential equations driven by Lévy processes.The coefficients of the equations are random functions depending on time and space variables,and no smoothness assumption of the coefficients is assumed.     

Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.     

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

A novel collocation method based on Genocchi wavelet is presented for the numerical solution of fractional differential equations and time‐fractional partial differential equations with delay. In this work, to achieve the approximate solution with height accuracy, we employed the operational matrix of integer derivative and the pseudo‐operational matrix of fractional derivative in Caputo sense. Also, based on Genocchi function properties, we presented delay and pantograph operational matrices of Genocchi wavelet functions (GWFs). Due to operational and pseudo‐operational matrices, the equations under this study can be turned into nonlinear algebraic equations with the unknown GWF coefficients. For illustrating the upper bound of error for the proposed method, we estimate the error in the sense of Sobolev space. In addition, to demonstrate the efficacy of the pseudo‐operational matrix of fractional derivative, we investigate the upper bound of error for the mentioned matrix. Finally, the algorithm based on the proposed approach is implemented for some numerical experiments to confirm accuracy and applicability.     

A Feynman-Kac type formula for a fixed delay CIR model

Stochastic delay differential equations (SDDE’s) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fixed delay CIR process. Moreover, for the fixed delay CIR process, we derive a Feynman-Kac type formula, leading to a generalized exponential-affine formula, which is used to determine a bond pricing formula when the interest rate follows the delay’s equation. It turns out that, for each maturity time T, the instantaneous forward rate is an affine function (with time dependent coefficients) of the rate process and of an auxiliary process (also depending on T). The coefficients satisfy a system of deterministic differential equations.     

Type II hidden symmetries through weak symmetries for some wave equations

The Type II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie-point symmetry. In [Gandarias RML. Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations. J Math Anal Appl 2008;348:752–9] it was shown that the provenance of the Type II Lie point hidden symmetries found for differential equations can be explained by considering weak symmetries or conditional symmetries of the original PDE.In this paper we analyze the connection between one of the methods analyzed in [Abraham-Shrauner B, Govinder KS. Provenance of Type II hidden symmetries from nonlinear partial differential equations. J Nonlin Math Phys 2006;13:612–22] and the weak symmetries of some partial differential equations in order to determine the source of these hidden symmetries. We have considered some of the models presented in [Abraham-Shrauner B, Govinder KS. Provenance of Type II hidden symmetries from nonlinear partial differential equations. J Nonlin Math Phys 2006;13:612–22], as well as the linear two-dimensional and three-dimensional wave equations [Abraham-Shrauner B, Govinder KS, Arrigo JA. Type II hidden symmetries of the linear 2D and 3D wave equations. J h Phys A Math Theor 2006;39:5739–47].     

An L p -theory of Stochastic PDEs of Divergence Form on Lipschitz Domains

Stochastic partial differential equations of divergence form are considered on Lipschitz domains. Existence and uniqueness results are given in weighted Sobolev spaces. It is allowed that the coefficients of the equations substantially oscillate or blow up near the boundary.