Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

     

Asymptotic behaviour and zeros of solutions of linear differential equations

Summary. Certain classes of linear differential equations are investigated for which the distribution of zeros of their solutions determines their asymptotic behaviour. These results generalize those already obtained for the second order linear differential equations to equations of arbitrary order.     

One method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid

We propose a simple algebraic method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid. The problem reduces to consecutively solving three linear partial differential equations for a nonviscous fluid and to solving three linear partial differential equations and one first-order ordinary differential equation for a viscous fluid. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147,No. 1, pp. 64–72, April, 2006.     

On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.     

Application of the method of special series to the representation of solutions of the Lin-Reissner-Tsien equation

For the two-dimensional Lin-Reissner-Tsien equation, which describes nonstationary gas flows, we construct new classes of solutions with functional arbitrariness in the form of series in powers of specially chosen functions. Coefficients of such series are found successively as solutions of linear ordinary differential equations or as solutions of linear partial differential equations. The use of special series whose coefficients are determined by linear partial differential equations allowed us to satisfy two given additional boundary conditions exactly. For one class of flows, these coefficients were found in an explicit form from linear equations of the hyperbolic type; for another one, they were found from linear equations of the parabolic type. This circumstance was used to prove the convergence of such series and to study the asymptotics of the solutions constructed. We present results of numerical calculations on nonstationary transonic flow around a wedge.     

Unicity of meromorphic solutions of partial differential equations

In this survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Jacobian polynomials. We prove a uniqueness theorem of meromorphic functions of several complex variables sharing three values taking into account multiplicity such that one of the meromorphic functions satisfies a nonlinear partial differential equations of the first order with meromorphic coefficients, which extends the Brosch??s uniqueness theorem related to meromorphic solutions of nonlinear ordinary differential equations of the first order.     

Bivariate delta-evolution equations and convolution polynomials: Computing polynomial expansions of solutions

This paper describes an application of Rota and collaborator’s ideas, about the foundation on combinatorial theory, to the computing of solutions of some linear functional partial differential equations. We give a dynamical interpretation of the convolution families of polynomials. Concretely, we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series. This is the starting point to get symbolic solutions for some functional-partial differential equations. We introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional-evolution equations related to delta-operators. We put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations. We create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results.     

Approximated solutions in rational form for systems of differential equations

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Padé approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method

The purpose of the present paper is to introduce a method, probably for the first time, to predict the multiplicity of the solutions of nonlinear boundary value problems. This procedure can be easily applied on nonlinear ordinary differential equations with boundary conditions. This method, as will be seen, besides anticipating of multiplicity of the solutions of the nonlinear differential equations, calculates effectively the all branches of the solutions (on the condition that, there exist such solutions for the problem) analytically at the same time. In this manner, for practical use in science and engineering, this method might give new unfamiliar class of solutions which is of fundamental interest and furthermore, the proposed approach convinces to apply it on nonlinear equations by today’s powerful software programs so that it does not need tedious stages of evaluation and can be used without studying the whole theory. In fact, this technique has new point of view to well-known powerful analytical method for nonlinear differential equations namely homotopy analysis method (HAM). Everyone familiar to HAM knows that the convergence-controller parameter plays important role to guarantee the convergence of the solutions of nonlinear differential equations. It is shown that the convergence-controller parameter plays a fundamental role in the prediction of multiplicity of solutions and all branches of solutions are obtained simultaneously by one initial approximation guess, one auxiliary linear operator and one auxiliary function. The validity and reliability of the method is tested by its application to some nonlinear exactly solvable differential equations which is practical in science and engineering.     

Lump solutions to a generalized Hietarinta-type equation via symbolic computation

Lump solutions are one of important solutions to partial differential equations, both linear and nonlinear. This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms. The key is a Hirota bilinear form. Lump solutions are constructed via symbolic computations with Maple, and specific reductions of the resulting lump solutions are made. Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented, together with three-dimensional plots and density plots of the lump solutions.     

Mean value properties of solutions of linear partial differential equations

The present paper suggests a uniform viewpoint to mean value theorems for linear elliptic and hyperbolic partial differential equations that, in a certain cases, allows one to obtain new mean value formulas. Also, the authors consider a method for obtaining mean value formulas for elliptic equations from those for hyperbolic equations. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

On zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

Linear relations among algebraic solutions of differential equations

For any univariate polynomial with coefficients in a differential field of characteristic zero and any integer, q, there exists an associated nonzero linear ordinary differential equation (LODE) with the following two properties. Each term of the LODE lies in the differential field generated by the rational numbers and the coefficients of the polynomial, and the qth power of each root of the polynomial is a solution of this LODE. This LODE is called a qth power resolvent of the polynomial. We will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the αth power resolvent of the polynomial, where α is an indeterminate that takes the place of q. We will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives. We will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the (nondifferential) Galois group. Finally, we will use a differential resolvent to solve the Riccati equation.     

On existence of solutions of differential‐difference equations

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non‐linear differential‐difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential‐difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich‐type theorem and Malmquist‐type theorem about differential equations to the case of differential‐difference equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Homogeneous linear differential equations with only periodic solutions

We determine all third order homogeneous linear differential equations with periodic coefficients and only periodic solutions. The method extends tonth order equations. As an application, we show that the Laguerre-Forsyth canonical form cannot be used for global investigations i projective differential geometry. Research supported by NSF Grant GP-8176. To the Memory of Eri Jabotinsky     

Asymptotic solutions and error estimates for linear systems of difference and differential equations

Classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant, i.e., k+o(1) as t tends to infinity. Here we are interested in more precise information about the o(1) terms, specifically how they depend precisely on certain perturbation terms in the equation. Results along these lines were given by Gel'fond and Kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations.     

Controlling solutions to a linear differential equation

Both global attainability and global reducibility problems for a controllable system equivalent to a linear differential equation are positively solved. Furthermore, the existence of a linear equation having a given Cauchy matrix on a given segment and coinciding with given equations from the left and right of that segment is proved. The results obtained allow one to construct a linear equation with the fundamental solutions system possessing preassigned properties.     

Representation of solutions and stability of linear differential-difference equations in a Banach space

In this paper the theory of linear delay differential equations is extended in three directions. One, the underlying phase space is allowed to be a Banach space so that equations with unbounded operators may be considered. Two, the delay is permitted to be effective over an infinite interval and a connection is made between this type of system and neutral systems whose delay is effective over a finite interval. Three, a theory of uniform asymptotic stability for linear delay differential equations in a Hilbert space is developed.     

Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We develop index theories for linear selfadjoint operator equations and investigate multiple solutions for asymptotically linear operator equations. The operator equations consist of two kinds: the first has finite Morse index and can be used to investigate second order Hamiltonian systems and elliptic partial differential equations; the second may have infinite Morse index and can be used to investigate first order Hamiltonian systems.