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1.
We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed
for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α)
and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These
are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations
is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property,
namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation,
additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate
the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types
of equations in a space with any number of dimensions.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 370–377, March, 2009. 相似文献
2.
T. A. Burton 《Annali di Matematica Pura ed Applicata》1973,95(1):193-209
Summary In this paper we consider a system of two first order differential equations {x′=P(x,y). y′=Q(x,y)}. We usually assume that
∂P/∂x+∂Q/∂y vanish identically in a certain region. A number of conditions are then given to insure boundedness of solutions or asymptotic
stability of the zero solution.
Entrata in Redazione il 5 gennaio 1972. 相似文献
3.
Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument 总被引:1,自引:0,他引:1
J. CABALLERO B. LOPEZ K. SADARANGANI 《数学学报(英文版)》2007,23(9):1719-1728
We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x(t) = a(t) + (Tx)(t) ∫0^σ(t) u(t, s, x(s), x(λs))ds 0 〈 λ 〈 1 which can be considered in connection with the following Cauchy problem x'(t) = u(t, s, x(t), x(λt)), t ∈ [0, 1], 0 〈 λ 〈 1 x(0) = u0. 相似文献
4.
ZHANG Fubao 《中国科学A辑(英文版)》2001,44(5):631-644
In this paper, we consider the higher dimensional second order differential equations of the formẍ + ∇F(x,t) = 0,x ∈R
n
with a class of weakly coupled potentials F( x, t ), periodically depending on t. We prove the existence of infinitely many
quasi-periodic solutions for such equations via the KAM theorem. 相似文献
5.
David E. Dobbs 《International Journal of Mathematical Education in Science & Technology》2013,44(5):748-752
This self-contained note could find classroom use in a course on differential equations. It is proved that if y1(x) and y2(x) are C 2 -functions whose Wronskian is never zero for α < x < β, then y1 and y2 form a fundamental solution set for a uniquely determined second-order linear homogeneous ordinary differential equation, y″ + p(x)y′ + q(x)y = 0, whose coefficients, p(x) and q(x), are continuous on (α, β). 相似文献
6.
袁荣 《应用数学学报(英文版)》1998,14(1):68-73
Itiswellknownthattheexistenceofalmostperiodicsolutionsiscloselyrelatedtothestabilityofsolutions.Forfunctionaldifferentialequationswithinfinitedelay,Y.Hin.[5'6]studiedtheproblemsontheexistenceofalmostperiodicsolutionsandthestability.However,therearefewpapersll2]dealingwithneutralfunctionaldifferentialequationswithinfinitedelay.Inthepresentpaper,forneutralfunctionaldifferentialequationswithinfinitedelay,weprovetheinherencetheoremfortheuniformlystableoperatorD(t),definethestabilitywithrespecttot… 相似文献
7.
Shige Peng 《Probability Theory and Related Fields》1999,113(4):473-499
We have obtained the following limit theorem: if a sequence of RCLL supersolutions of a backward stochastic differential
equations (BSDE) converges monotonically up to (y
t
) with E[sup
t
|y
t
|2] < ∞, then (y
t
) itself is a RCLL supersolution of the same BSDE (Theorem 2.4 and 3.6).
We apply this result to the following two problems: 1) nonlinear Doob–Meyer Decomposition Theorem. 2) the smallest supersolution
of a BSDE with constraints on the solution (y, z). The constraints may be non convex with respect to (y, z) and may be only measurable with respect to the time variable t. this result may be applied to the pricing of hedging contingent claims with constrained portfolios and/or wealth processes.
Received: 3 June 1997 / Revised version: 18 January 1998 相似文献
8.
We show that a finite generalized polygon Γ is Moufang with respect to a groupG if and only if for every flag {x, y} of Γ, the subgroupG
1(x, y) ofG fixing every element incident with one ofx, y acts transitively on the set of apartments containing the elementsu, x, y, w, whereu≠y (resp.w≠x) is an arbitrary element incident withx (resp.y).
Research Associate at the National Fund of Scientific Research of Belgium.
Research partially supported by NSF Grant DMS-8901904. 相似文献
9.
Let 2 ≤ p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [27], [2], [3], [5], [23].
Received: 25 April 2008 相似文献
10.
K. E. Swick 《Annali di Matematica Pura ed Applicata》1977,114(1):1-26
Summary Solutions of
are said to converge if every pair of solutions x(t), y(t) satisfy x(t) − y(t) →0 as t → ∞. An invariance principle of LaSalle is used to determine conditions under which the solutions of
converge. In certain cases the approach used does not require boundedness of solutions as has been required in most previous
results on convergence of solutions. The results of this investigation are applied to a number of nonlinear second order differential
equations. Sufficient conditions are also found for the convergence of solutions of certain functional differential equations.
Entrata in Redazione il 10 febbraio 1976. 相似文献
11.
Adrian Constantin 《Annali dell'Universita di Ferrara》1995,41(1):1-4
LetH be a complex Hilbert space and letB be the space of all bounded linear operators fromH intoH with the strong operator topology. We will give a boundedness result for the solutions of the differential equationx′=A(t)x+f(t,x) whereA: I=[t
0, ∞)→B is continuous,f: I×H→H is also continuous and for every bounded setS⊂I×H there exists a constantM(S)>0 such that |f(t,x)−f(t,y)|≤M(S)|x−y|,(t,x), (t,y)∈S.
Sunto SiaH uno spazio di Hilbert complesso e siaB lo spazio degli operatori lineari limitati daH inH, con la topologia forte. In questo lavoro si prova un risultato di limitatezza per le soluzioni dell'equazione differenzialex′=A(t)x+f(t,x), doveA: I=[t 0, ∞)→B è continua,f: I×H→H è continua e per ogni insieme limitatoS⊂I×H esiste una costanteM(S)>0 tale che |f(t,x)−f(t,y)|≤M(S)|x−y| per ogni(t,x), (t,y)∈S.相似文献
12.
Andreas Dalcher 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1956,7(4):273-304
Summary Stochastic processes of the following type are considered. At random time points, the variablex(t) jumps fromy tox, say. The heightsx–y of the jumps have a given distributionG
*(x–y) that may depend ony ort. Between the jumps,x(t) is a solution to a given differential equationdx/dt=x(x, t). We look for the distributionF(x, t) ofx at timet>0,F(x, 0) being given. In the stationary case, stable distributions are investigated.If there is a lower boundaryx
0 and ifF(x
0)>0, the problem is similar to the queueing problem. We solve it in the stationary case with integral equations of the Volterra type. Other problems can be transformed to differential equations for the moment generating functions. These equations are partial in the non stationary and ordinary in the stationary case. 相似文献
13.
Soon-Mo Jung 《数学学报(英文版)》2010,26(11):2085-2092
We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g(x, y) + h(x, y)y' =0. 相似文献
14.
The global behaviour of the control systems described by thepair of differential equations x{dot} = f(x)±g(y)+p(t), y{dot} = f(x)±g(y)+p(t) has been investigated. Here f(x), g(y), h(x) and k(y) are polynomialsof odd degree with leading coefficients positive and p(t) andq(t) are bounded functions of time. Sufficient conditions havebeen found under which the trajectories of the above systemmay eventually be confined in a subset of (x, y, t)-space, thusgiving bounds on the amplitude of periodic as well as aperiodicoscillations. Further bounds on the amplitude of oscillationshave been investigated by finding regions in (x,y,t)-space fromwhich all trajectories eventually leave and into which no trajectoriesenter. Thus sufficient conditions have been derived for theexistence of an annulus in which oscillatory behaviour may beconfined. 相似文献
15.
V. A. Strakhov 《Mathematical Notes》1977,21(2):85-90
For the two operatorsLy=y
n
+σ
k=0
n−2
p
k
(x)y(
k
) and Ry=yn+σ
k=0
n−2
pk(x)y(k) with a common set of boundary conditions we establish a connection between pk(x) and Pk(x) in the case where the weight numbers coincide and a finite number of the eigenvalues do not coincide, in terms of the
eigenfunctions of these operators corresponding to the noncoincident eigenvalues and the derivatives of these functions. This
enables us to recover the operator L from the operator R by solving a system of nonlinear ordinary differential equations.
For Sturm-Liouville operators an analogous relation is proved for the case where infinitely many eigenvalues do not coincide.
Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 151–160, February, 1977.
I wish to express my thanks to my scientific adviser V. A. Sadovnich. 相似文献
16.
A. I. Gerko 《Mathematical Notes》2000,67(6):707-717
In the paper methods from the theory of extensions of dynamical systems are used to studyβ-differential equations whose solutions possess the uniqueness property and depend continuously on the initial data and on
the right-hand side of the equation. The Zhikov-Bronshtein theorems concerning asymptotically almost periodic solutions of
ordinary differential equations are extended toβ-differential equations (in particular, to total differential equations). Along with asymptotic almost periodicity, we also
consider asymptotic recurrence, weak asymptotic distality, and asymptotic distality. To the equations we associate dynamical
systems generated by the space of the right-hand sides and the spaces of the solutions and of the initial data of solutions
of the equation. Generally, the phase semigroups of the dynamical systems are not locally compact.
Translated fromMatermaticheskie Zametki, Vol. 67, No. 6, pp. 837–851, June, 2000. 相似文献
17.
In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If
φ(x) is a polynomial, then φ(x) is called a polynomial solution. 相似文献
18.
Let T = T(p, q, α) be the number of solutions of the congruence xα ≡ 1 (mod pηqθ). Let A
and B be sets of primes satisfying x1 < p ≤ x2 and y1 < q ≤ y2, respectively. A mean value estimation
of
is given.
Supported by National Natural Science Foundation of China (No. 19971024) and Zhejiang Provincial Natural
Science Foundation of China (No. 199047) 相似文献
19.
We consider the initial boundary-value problem for the quasilinear diffusion equation ∂
t
u+u − DΔu = K(x, y)(1 + γ(u + ϕ(x, y))) describing the dynamics of optical systems with controlled feedback wave intensity modulation K(x, y) in the presence of incoming-wave phase perturbations ϕ(x, y). The control problem for the parameter K(x, y) is formulated with the objective of smoothing out the spatial nonhomogeneities of the total output phase u(x, y, T) + ϕ(x, y). We prove existence and uniqueness theorems for the generalized solutions of the direct and conjugate problems, solvability
theorems for the optimization problems, and Frechet-differentiability of the objective functional. A formula for the functional
gradient is derived and the efficiency of the gradient projection method is demonstrated numerically.
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Translated from Prikladnaya Matematika i Informatika, No. 20, pp. 80–99, 2005. 相似文献
20.
T. N. Shorey 《Proceedings Mathematical Sciences》1984,93(2-3):109-116
For given positive integersa andb, the equationa(x + 1)… (x + k) =b(y+1)… (y + k) in positive integers is considered. More general equations are also considered. 相似文献