Hamiltonian and algebro-geometric integrals of stationary equations of KdV type

1980 ◽  
Vol 87 (2) ◽  
pp. 295-305 ◽  
Author(s):  
George Wilson
Keyword(s):  
Unknown Function ◽  
Evolution Equations ◽  
Differential Operators ◽  
Schrodinger Operator ◽  
Order Zero ◽  
Kdv Equations ◽  
Ordinary Differential Operators ◽  
De Vries ◽  
Image Position ◽  
The Right

In this paper we shall generalize a theorem of Bogoyavl'enskii (2) showing the equivalence of two apparently different families of integrals of the ‘higher stationary Korteweg–de Vries (KdV)’ equations. We recall that the KdV equations form a hierarchy of evolution equationsfor an unknown function u(x, t). The equations can be written in ‘Lax form’ (7)where L is the Schrödinger operator (δ2/δx2) + u, and P+ runs through all the (ordinary) differential operators such that the commutator on the right of (1·1) has order zero.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

An integral inequality with an application to ordinary differential operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 35-44 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Integral Inequality ◽  
Differential Operators ◽  
Linear Manifold ◽  
Integral Inequalities ◽  
Compact Interval ◽  
Ordinary Differential Operators ◽  
Image Position

SynopsisThis paper is concerned with integral inequalities of the formwhere p, q are real-valued coefficients, with p and w non-negative, on the compact interval [a, b] and D is a linear manifold of functions so chosen that all three integrals are absolutely convergent.

Commuting flows and conservation laws for Lax equations

1979 ◽  
Vol 86 (1) ◽  
pp. 131-143 ◽  
Author(s):  
George Wilson
Keyword(s):  
Evolution Equations ◽  
Differential Operators ◽  
Kdv Equation ◽  
Lax Representation ◽  
Linear Evolution ◽  
Commuting Flows ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

In recent years there has been great progress in the study of certain systems of non-linear partial differential equations, namely those that have a ‘Lax representation’Here P and L are linear differential operators in one variable x, whose coefficients are l × l matrices of functions of x and t. Thus L has the formwhere each ui is a matrix of functions ui,αβ(x, t), 1 ≤ α, β ≤ l. The symbol Lt means that we differentiate each coefficient of L, and as usual [P, L] = PL − LP. The coefficients of P are supposed to be polynomials in the ui, αβ and their x-derivatives, so that (1·1) is equivalent to a system of non-linear ‘evolution equations’ for the variables ui, αβ. The simplest example is the Korteweg–de Vries (KdV) equationwhich has a Lax representation with(Here l = 1, and there is only one coefficient ui, αβ. ) The connexion between the KdV equation and this ‘Schrodinger operator’ L was discovered by Gardner, Greene, Kruskal and Miura(6), but it was P. Lax (13) who first pointed out explicitly what we call the Lax representation given by (1·1) and (1·2). The notation (P, L), due to Gel'fand and Dikii(7), reflects this fact.

Limit Point Criteria for Differential Equations

1972 ◽  
Vol 24 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Limit Point ◽  
Differential Operators ◽  
Central Importance ◽  
Linearly Independent ◽  
Ordinary Differential Operators ◽  
Image Position

For certain ordinary differential operators L of order 2n, this paper considers the problem of determining the number of linearly independent solutions of class L2[a, ∞) of the equation L(y) = λy. Of central importance is the operator0.1where the coefficients pi are real. For this L, classical results give that the number m of linearly independent L2[a, ∞) solutions of L(y) = λy is the same for all non-real λ, and is at least n [10, Chapter V]. When m = n, the operator L is said to be in the limit-point condition at infinity. We consider here conditions on the coefficients pi of L which imply m = n. These conditions are in the form of limitations on the growth of the coefficients.

Estimates for the state density for ordinary differential operators with white gaussian noise potential

1983 ◽  
Vol 95 (3-4) ◽  
pp. 263-274
Author(s):  
V. B. Moscatelli ◽  
M. Thompson
Keyword(s):  
Wiener Process ◽  
Lower Bounds ◽  
Gaussian Noise ◽  
Differential Operators ◽  
Asymptotic Properties ◽  
State Density ◽  
The State ◽  
Upper And Lower Bounds ◽  
Ordinary Differential Operators ◽  
Image Position

SynopsisThe present paper is concerned with developing the existence and asymptotic properties of the state density N(λ) associated with certain higher order random ordinary differential operators A of the formwhere Ao has homogeneous and ergodic coefficients with respect to the σ-algebra generated by the Wiener process q(ω, x). The analysis uses the Weyl min-max principle to determine rough upper and lower bounds for N(λ).

scholarly journals Two Kinds of New Integrable Couplings of the Negative-Order Korteweg-de Vries Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Binlu Feng ◽  
Yufeng Zhang
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Loop Algebras ◽  
Negative Order ◽  
Integrable Couplings ◽  
De Vries ◽  
Integrable Coupling

Based on some known loop algebras with finite dimensions, two different negative-order integrable couplings of the negative-order Korteweg-de Vries (KdV) hierarchy of evolution equations are generated by making use of the Tu scheme, from which the corresponding negative-order integrable couplings of the negative-order KdV equations are followed to be obtained. The resulting Hamiltonian structure of one negative integrable coupling is derived from the variational identity.

Perturbation of the Continuous Spectrum of Systems of Ordinary Differential Operators

1962 ◽  
Vol 14 ◽  
pp. 359-378 ◽  
Author(s):  
John B. Butler
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Eigenvalue Problem ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Integrable Solution ◽  
Bounded Interval ◽  
Ordinary Differential Operators ◽  
Piecewise Continuous ◽  
Image Position

Letbe an ordinary differential operator of order h whose coefficients are (η, η) matrices defined on the interval 0 ≤ x < ∞, hη = n = 2v. Let the operator L0 be formally self adjoint and let v boundary conditions be given at x = 0 such that the eigenvalue problem(1.1)has no non-trivial square integrable solution. This paper deals with the perturbed operator L∈ = L0 + ∈q where ∈ is a real parameter and q(x) is a bounded positive (η, η) matrix operator with piecewise continuous elements 0 ≤ x < ∞. Sufficient conditions involving L0, q are given such that L∈ determines a selfadjoint operator H∈ and such that the spectral measure E∈(Δ′) corresponding to H∈ is an analytic function of ∈, where Δ′ is a subset of a fixed bounded interval Δ = [α, β]. The results include and improve results obtained for scalar differential operators in an earlier paper (3).

Residual Gas Reactions in the Electron Microscope: I. Qualitative Observations on the Water Gas Reaction

1968 ◽  
Vol 26 ◽  
pp. 292-293
Author(s):  
Richard E. Hartman ◽  
Roberta S. Hartman ◽  
Peter L. Ramos
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Gas Phase ◽  
Absolute Temperature ◽  
Residual Gas ◽  
Gas Reactions ◽  
Image Position ◽  
The Right ◽  
Water Gas ◽  
Thinning Rate

The action of water and the electron beam on organic specimens in the electron microscope results in the removal of oxidizable material (primarily hydrogen and carbon) by reactions similar to the water gas reaction .which has the form:The energy required to force the reaction to the right is supplied by the interaction of the electron beam with the specimen.The mass of water striking the specimen is given by:where u = gH2O/cm2 sec, PH2O = partial pressure of water in Torr, & T = absolute temperature of the gas phase. If it is assumed that mass is removed from the specimen by a reaction approximated by (1) and that the specimen is uniformly thinned by the reaction, then the thinning rate in A/ min iswhere x = thickness of the specimen in A, t = time in minutes, & E = efficiency (the fraction of the water striking the specimen which reacts with it).

scholarly journals Approximate Analytical Solution of Nonlinear Evolution Equations

2020 ◽  
Author(s):  
Laxmikanta Mandi ◽  
Kaushik Roy ◽  
Prasanta Chatterjee
Keyword(s):  
Acoustic Waves ◽  
Evolution Equations ◽  
Solitary Wave Solution ◽  
Perturbation Technique ◽  
Nonlinear Evolution Equations ◽  
Charged Dust ◽  
De Vries ◽  
Frame Work ◽  
Dust Grain ◽  
Charged Ions

Analytical solitary wave solution of the dust ion acoustic waves (DIAWs) is studied in the frame-work of Korteweg-de Vries (KdV), damped force Korteweg-de Vries (DFKdV), damped force modified Korteweg-de Vries (DFMKdV) and damped forced Zakharov-Kuznetsov (DFZK) equations in an unmagnetized collisional dusty plasma consisting of negatively charged dust grain, positively charged ions, Maxwellian distributed electrons and neutral particles. Using reductive perturbation technique (RPT), the evolution equations are obtained for DIAWs.

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