In a 1977 paper of McCoy, Tracy and Wu [2] there appeared for the first time the solution of a Painlevé equation in terms of Fredholm determinants of integral operators. Specifically, it was shown that a one-parameter family of solutions of the equation ψ ′′ (t) + t −1 ψ ′ (t) = 1 2 sinh 2ψ + 2α t −1 sinh ψ, (1) a special case of the Painlevé III equation, is given by ψ(t) =

In this work, we successfully construct the new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. The idea introduced in this paper can be applied to other nonlinear evolution equations.

We consider an eigenvalue problem for a system in [0, 1]:        0 1 1 0 d dx + p11(x) p12(x) p21(x) p22(x) ϕ (1) (x) ϕ (2) (x) = λ ϕ (1) (x) ϕ (2) (x) ϕ (2) (0) cosh µ − ϕ (1) (0) sinh µ = ϕ (2) (1) cosh ν + ϕ (1) (1) sinh ν = 0 with constants µ, ν ∈ C. Under the assumption that p21, p22 are known, we prove a uniqueness theorem and provide a reconstruction formula for p11 and p12 from t...

We show that noncompact simply connected harmonic manifolds with volume density Θ p (r) = sinh n−1 r is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density Θ p (r) = sinh 2n−1 r cosh r is isometric to the complex hyperbolic space. A similar result is also proved for Quaternionic Kähler manifolds. Using our methods we get an alterna...

In this manuscript we present a detailed investigation of the form factors of boundary fields of the sinh-Gordon model with a particular type of Dirichlet boundary condition, corresponding to zero value of the sinh-Gordon field at the boundary, at the self-dual point. We follow for this the boundary form factor program recently proposed by Z. Bajnok, L. Palla and G. Takacs [1], extending the an...

The relationship between solutions of the sinh-Laplace equation and the determination of various kinds of surfaces of constant Gaussian curvature, both positive and negative, will be investigated here. It is shown that when the metric is given in a particular set of coordinates, the Gaussian curvature is related to the sinh-Laplace equation in a direct way. The fundamental equations of surface ...

and Applied Analysis 3 (a, b), and let g󸀠(x) ̸ = 0 on (a, b). If f󸀠(x)/g󸀠(x) is increasing (decreasing) on (a, b), then so are f (x) − f (a) g (x) − g (a) , f (x) − f (b) g (x) − g (b) . (11) If f󸀠(x)/g󸀠(x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 6 (see [11, Lemma 1.1]). Suppose that the power series f(x) = ∑ ∞ n=0 a n x n and g(x) = ∑∞ n=0 b n x n hav...

We have calculated the energy per site for the ground state of antiferromagnetic quantum spin chain with variable range exchange h(j − k) ∝ sinh a sinh a(j − k) in the framework of the asymptotic Bethe ansatz. By expanding it in powers of e, we have confirmed the value of the second-neighbor correlator for the model with nearest-neighbor exchange obtained earlier in the atomic limit of the Hubb...

In [5] and [2] the authors describe the remarkable effectiveness of the doubly exponential ‘tanh-sinh’ transformation for numerical integration —even for quite unruly integrands. Our intention in this note is to provide a theoretical underpinning when the integrand is analytic for the observed superlinear convergence of the method. Our analysis rests on the corresponding but somewhat easier ana...

Using the Quantum Inverse Scattering Method for the XXZ model with open boundary conditions, we obtained the determinant formula for the six vertex model with reflecting end. E-mail address: [email protected] 1 1 Introducntion Determinant representations of correlation functions of one dimensional quantum integrable models were studied extensively[1]. Especially Essler et. al. studie...