Discrete dressing transformations and Painlevé equations

We propose a discrete analog of the dressing transformation. Our starting point is a variant of the quotient-difference algorithm which, in this case, corresponds to a linear problem with shifts in the eigenvalues. The proper periodicity conditions lead to one-dimensional systems which are discrete Painlevé equations. We obtain thus the alternate d-P_II equation and a novel form for the discrete P_IV equation.     

Riccati-type Bcklund transformations of nonisospectral and generalized variable-coefficient KdV equations

We extend the method of constructing Bcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg–de Vries(KdV) equations as examples, their B¨acklund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Especially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.     

New Bäcklund transformations for the third and fourth Painlevé equations to equations of second order and higher degree

We give new Bäcklund transformations for the third and fourth Painlevé equations, to equations of second order and higher degree.     

Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and of non-linear Schrödinger equations

We consider, in a 1+3 space time, arbitrary (finite) systems of nonlinear Klein-Gordon equations (respectively Schrödinger equations) with an arbitrary local and analytic non-linearity in the unknown and its first and second order space-time (respectively first order space) derivatives, having no constant or linear terms. No restriction is given on the frequency sign of the initial data. In the case of non-linear Klein-Gordon equations all masses are supposed to be different from zero.We prove, for such systems, that the wave operator (fromt= tot=0) exists on a domain of small entire test functions of exponential type and that the analytic Cauchy problem, in ⁺׳, has a unique solution for each initial condition (att=0) being in the image of the wave operator. The decay properties of such solutions are discussed in detail.Partially supported by the Swiss National Science FoundationOn leave from Institut de Physique Théorique, 32 Bd d'Ivoy, CH-1211 Geneve 4 Switzerland.     

Existence of strong Bäcklund transformations in four or more dimensions and generalization of a family of Bäcklund transformations

A Bäcklund transformation between two (sets of) differential equations is strong if the transformation equations already imply the two equations. For each dimension n=2 ^k , k1, the existence of such strong transformations is proved by constructing a wide variety of them. A simple generalization of a known family of Bäcklund transformations is also given. One such provides a useful analogy for Yang's instanton equations.     

On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy.     

Complex Langevin equations and Schwinger–Dyson equations

Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger–Dyson equations of the associated quantum field theory. Specific examples in zero dimensions and on a lattice are given. The relevance to the study of quantum field theory solution space is discussed.     

Form invariance and Lie symmetry of equations of non—holonomic systems

In this paper,we study the relation between the form invariance and Lie symmetry of non-holonomic systems.Firstly,we give the definitions and criteria of the form invariance and Lie symmetry in the systems.Next,their relation is deduced.We show that the structure equation and conserved quantity of the form invariance and Lie symmetry of non-holonomic systems have the same form.Finally,we give an example to illustrate the application of the result.     

Uniform boundedness of conditional gauge and Schrödinger equations

We prove that for a bounded domainD ?R ⁿ withC ² boundary and \(q \in K_n^{loc} (n \geqq 3) if E^x \exp \int\limits_0^{\tau _D } {q(x_t )dt} \mathop \ddag \limits_--- \infty \) inD, then $$\mathop {\sup }\limits_{\mathop {x \in D}\limits_{z \in \partial D} } E_z^x \exp \int\limits_0^{\tau _D } {q(x_t )dt}< + \infty $$ ({x _t : Brownian motion}) The important corollary of this result is that if the Schrödinger equation Δ/2u+qu=0 has a strictly positive solution onD, then for anyD ₀ ? ?D, there exists a constantC=C(n,q,D,D ₀) such that for anyf εL ¹(?D, σ), (σ: area measure on ?D) we have $$\mathop {\sup |}\limits_{x \in D_0 } u_f (x)| \mathop< \limits_ = C\int\limits_{\partial D} {|f(y)|\sigma (dy)} $$ whereu _f is the solution of the Schrödinger equation corresponding to the boundary valuef. To prove the main result we set up the following estimate inequalities on the Poisson kernelK(x,z) corresponding to the Laplace operator: $$C_1 \frac{{d(x,\partial D)}}{{|x - z|^n }}\mathop< \limits_ = K(x,z)\mathop< \limits_ = C_2 \frac{{d(x,\partial D)}}{{|x - z|^n }},x \in D,z \in \partial D$$ whereC ₁ andC ₂ are constants depending onn andD.     

Current-driven transformations of a skyrmion tube and a bobber in stepped nanostructures of chiral magnets

Magnetic skyrmion tubes and bobbers are two types of different nanoscale spin configurations that can coexist in nanostructures of chiral magnets.They are then proposed to be utilized as binary bits to build racetrack memory devices.The ability to manipulate the two magnetic objects controllably by current in nanostructures is the prerequisite to realize the device.Here,we demonstrate by numerical simulations that a magnetic bobber and a skyrmion tube can be transformed to each other using spinpolarized current in nanostripes with stepped shape.We also show such stepped nanostructures can be readily applied as the write head for the skyrmion-bobber-based racetrack memory.     

Hyper-Kähler metrics and a generalization of the Bogomolny equations

We generalize the Bogomolny equations to field equations on ^{3 n} and describe a twistor correspondence. We consider a general hyper-Kähler metric in dimension 4n with an action of the torusT ⁿ compatible with the hyper-Kähler structure. We prove that such a metric can be described in terms of theT ⁿ -solution of the field equations coming from the twistor space of the metric.     

Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev-Petviashvili equation

The infinite-dimensional symmetry group of the potential Kadomtsev-Petviashvili (PKP) equation is found and used to obtain a Bäcklund transformation, involving two arbitrary functions of time. This transformation is then used to generate several different types of solutions from the zero solution of the PKP equation.     

The operational solution of fractional-order differential equations,as well as Black–Scholes and heat-conduction equations

Operational solutions to fractional-order ordinary differential equations and to partial differential equations of the Black–Scholes and of Fourier heat conduction type are presented. Inverse differential operators, integral transforms, and generalized forms of Hermite and Laguerre polynomials with several variables and indices are used for their solution. Examples of the solution of ordinary differential equations and extended forms of the Fourier, Schrödinger, Black–Scholes, etc. type partial differential equations using the operational method are given. Equations that contain the Laguerre derivative are considered. The application of the operational method for the solution of a number of physical problems connected with charge dynamics in the framework of quantum mechanics and heat propagation is demonstrated.     

Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations

The fundamental solutionE(t,s,x,y) of time dependent Schrödinger equationsi?u/?t=?(1/2)Δu+V(t,x)u is studied. It is shown that

?E(t,s,x,y) is smooth and bounded fort≠s if the potential is sub-quadratic in the sense thatV(t,x)=o(|x|²) at infinity;

? in one dimension, ifV(t,x)=V(x) is time independent and super-quadratic in the sense thatV(x)≧C(1+|x|)^2+ε at infinity,C>0 and ε>0, thenE(t,s,x,y) is nowhereC ¹.

The result is explained in terms of the limiting behavior as the energy tends to infinity of the corresponding classical particle.     

A class of blowup and global analytical solutions of the viscoelastic Burgersʼ equations

In this Letter, by employing the perturbational method, we obtain a class of analytical self-similar solutions of the viscoelastic Burgers? equations. These solutions are of polynomial-type whose forms, remarkably, coincide with that given by Yuen for the other physical models, such as the compressible Euler or Navier–Stokes equations and two-component Camassa–Holm equations. Furthermore, we classify the initial conditions into several groups and then discuss the properties on blowup and global existence of the corresponding solutions, which may be readily seen from the phase diagram.     

Electron beam-induced structural transformations of MoO3 and MoO3?x crystalline nanostructures

Electron beam-induced damage and structural changes in MoO₃ and MoO_3−x single crystalline nanostructures were revealed by in situ transmission electron microscopy (TEM) examination (at 200 kV) after few minutes of concentrating the electron beam onto small areas (diameters between 25 and 200 nm) of the samples. The damage was evaluated recording TEM images, while the structural changes were revealed acquiring selected area electron diffraction patterns and high resolution transmission electron microscopy (HRTEM) images after different irradiation times. The as-received nanostructures of orthorhombic MoO₃ were transformed to a Magnéli’s phase of the oxide (γ-Mo₄O₁₁) after ~10 min of electron beam irradiation. The oxygen loss from the oxide promoted structural changes. HRTEM observations showed that, in the first stage of the reduction, oxygen vacancies generated by the electron beam are accommodated by forming crystallographic shear planes. At a later stage of the reduction process, a polycrystalline structure was developed with highly oxygen-deficient grains. The structural changes can be attributed to the local heating of the irradiated zone combined with radiolysis.     

A new family of discrete Painlevé equations and associated linearisable systems

We derive discrete systems which result from a second, not studied up to now, form of the q-P_VI equation. The derivation is based on two different procedures: “limits” and “degeneracies”. We obtain several new discrete Painlevé equations along with some linearisable systems. The parallel between the results for the standard form of q-P_VI and those of the new one is also established.