Real Analytic Solutions Research Articles

The Periodic Cauchy Problem for Novikov's Equation

We study the periodic Cauchy problem for an integrable equation with cubic nonlinearities introduced by V. Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, Novikov's equation has Lax pair representations and admits peakon solutions, but it has nonlinear terms that are cubic, rather than quadratic. We show the local well-posedness of the problem in Sobolev spaces and existence and uniqueness of solutions for all time using orbit invariants. Furthermore we prove a Cauchy-Kowalevski type theorem for this equation, that establishes the existence and uniqueness of real analytic solutions.

Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain

We address the question of well-posedness in spaces of analytic functions for the Cauchy problem for the hydrostatic incompressible Euler equations (inviscid primitive equations) on domains with boundary. By a suitable extension of the Cauchy–Kowalewski theorem we construct a locally in time, unique, real-analytic solution and give an explicit rate of decay of the radius of real-analyticity.

Generation of new classes of exactly solvable potential from the trigonometric Rosen-Morse potential

Exact analytic solutions of the Schrodinger equation are obtained for classes of newly constructed potentials which are generated from the trigonometric Rosen-Morse potential as the input reference potential via extended transformation method. A set of quantized energy spectra of the bound states and the corresponding wave functions of the generated potentials are obtained. We also focus on to the Romanovski Polynomials which is a family of the real orthogonal polynomials and is required to present exact real analytic solutions of the generated potentials.

Monge–Ampère foliations for degenerate solutions

We study the problem of the existence and the holomorphicity of the Monge–Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge–Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a positive answer to a question of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge–Ampère equation and we obtain a generalization the work of Wong on the classification of complete weighted circular domains.

A “Discretization” Technique for the Solution of ODEs II

A functional analytic technique was recently presented for finding discrete equivalent counterparts of initial value problems of ODEs and obtaining their real analytic solutions. In the current paper, this technique is extended to boundary value problems of ODEs and to the complex solutions of ODEs. In order to demonstrate this technique, it is applied to the classic Blasius problem of fluid mechanics. Apart from its real solution, its complex solution is also studied. The obtained results indicate that the complex Blasius function exhibits an oscillatory behavior and strengthen a conjecture regarding its singularities in the complex plane.

K‐symmetric AKS systems and flat immersions into spheres

We define a large class of integrable nonlinear PDEs, k‐symmetric AKS systems, with solutions that evolve on finite‐dimensional subalgebras of loop algebras and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the solution, and show that the problem of isometrically immersing n dimensional Euclidean space into a sphere of dimension 2n – 1 can be addressed via this scheme, producing infinitely many real analytic solutions.

Real analytic solutions for marginal deformations in open superstring field theory

We construct analytic solutions for marginal deformations satisfying the reality condition in open superstring field theory formulated by Berkovits when operator products made of the marginal operator and the associated superconformal primary field are regular. Our strategy is based on the recent observation by Erler that the problem of finding solutions for marginal deformations in open superstring field theory can be reduced to a problem in the bosonic theory of finding a finite gauge parameter for a certain pure-gauge configuration labeled by the parameter of the marginal deformation. We find a gauge transformation generated by a real gauge parameter which infinitesimally changes the deformation parameter and construct a finite gauge parameter by its path-ordered exponential. The resulting solution satisfies the reality condition by construction.

Real analytic zero solutions of linear partial differential operators with constant coefficients

It is shown that for a wide class of linear partial differential operators with constant coefficients the space of real analytic zero solutions does not admit a Schauder basis. This is based on results on the linear topological structure of the space of zero solutions and a careful analysis of the solvability with a real analytic parameter.

Truncated Microsupport and Hyperbolic Inequalities

AbstractWe prove that the k-truncated microsupport of the specialization of a complex of sheaves F along a submanifold is contained in the normal cone to the conormal bundle along the k-truncated microsupport of F. In the complex case, applying our estimates to , where is a coherent -module, we obtain new estimates for the truncated microsupport of real analytic and hyperfunction solutions. When is regular along Y we also obtain estimates for the truncated microsupport of the holomorphic solutions of the induced system along Y as well as for the nearby-cycle sheaf of when Y is a hypersurface.

Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations μ n (f) = b n for n ∈ N has a real analytic solution f on ω C R d for every right-hand side (b n ) n ∈ N ⊆ C and give a complete characterization of such sequences of analytic functionals (μ n ). We also show that every open set w C R d has a complex neighbourhood Ω C C d such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on Ω.

On the approximation of continuous functions by analytic solutions of universal functional-differential equations

The existence of an algebraic functional-differential equation P (y′(x), y′(x + log 2), …, y′(x + 5 log 2)) = 0 is proved such that the real-analytic solutions are dense in the space of continuous functions on every compact interval. A similar result holds for an algebraic functional-differential equation P(y′(x − 4πi), y′(x − 2πi), …, y′(x + 4πi)) = 0 (with i2 = −1), which is explicitly given: There are real-analytic solutions on the real line such that every continuous function defined on a compact interval can be approximated by these solutions with arbitrary accuracy.

Asymptotic expansions close to the singularity in Gowdy spacetimes

We consider Gowdy spacetimes under the assumption that the spatial hypersurfaces are diffeomorphic to the torus. The relevant equations are then wave map equations with the hyperbolic space as a target. In a paper by Grubiš;ić and Moncrief, a formal expansion of solutions in the direction towards the singularity was proposed. Later, Kichenassamy and Rendall constructed a family of real analytic solutions with the maximum number of free functions and the desired asymptotics at the singularity. The condition of real analyticity was subsequently removed by Rendall. In a previous paper, we proved that one can put a condition on initial data that leads to asymptotic expansions. However, control of up to and including three derivatives in L2 was necessary, and the condition was rather technical. The main point of the present paper is to demonstrate the existence of certain monotone quantities and to illustrate how these can be used to weaken the assumptions to one derivative in the sup norm. Furthermore, we demonstrate that the false spikes do not appear in the disc model. Finally, we show that knowledge concerning the behaviour of the solution (as time tends to the singularity) for one fixed spatial point in some situations can be used to conclude that there are smooth expansions in the neighbourhood of that spatial point.

Interpolation of Vector-Valued Real Analytic Functions

Let ω ⊆ Rd be an open domain. The sequentially complete DF-spaces E are characterized such that for each (some) discrete sequence (zn) ⊆ ω, a sequence of natural numbers (kn) and any family ( X n , α ) n ∈ N , | α | ⩽ k n ⊆ E the infinite system of equations ( ∂ | α | f ∂ z α ) ( z n ) = x n , α for n ∈ N , α ∈ N d , | α | ⩽ k n , has an E-valued real analytic solution f.

On the continuation of solutions for elliptic equations in two variables

We consider nonlinear elliptic differential equations of second order in two variables F(x,y,z(x,y),zx(x,y),…,zyy(x,y))=0, (x,y)∈Ω⊂R2. Supposing analyticity of F, we prove analyticity of the real solution z=z(x,y) in the open set Ω. Furthermore, we show that z may be continued as a real analytic solution for F=0 across the real analytic boundary arc Γ⊂∂Ω, if z satisfies one of the boundary conditions z=ϕ or zn=ψ(x,y,z,zt) on Γ with real analytic functions ϕ and ψ, respectively (zn denotes the derivative of z w.r.t. the outer normal n on Γ and zt its derivative w.r.t. the tangent). The proof is based on ideas of H. Lewy combined with a uniformization method. Studying quasilinear equations, we get somewhat better results concerning the initial regularity of the given solution and a little more insight.

Real analytic solutions of parabolic equations with time-measurable coefficients

We use Bernstein’s technique to show that for any fixed t t , strong solutions u ( t , x ) u(t,x) of the uniformly parabolic equation L u := a i j ( t ) u x i x j − u t = 0 Lu:=a^{ij}(t)u_{x_{i}x_{j}}-u_{t}=0 in Q Q are real analytic in Q ( t ) = { x : ( t , x ) ∈ Q } Q(t)=\{x:(t,x)\in Q\} . Here, Q ⊂ R d + 1 Q\subset \mathbb {R}^{d+1} is a bounded domain and the coefficients a i j ( t ) a^{ij}(t) are measurable. We also use Bernstein’s technique to obtain interior estimates for pure second derivatives of solutions of the fully nonlinear, uniformly parabolic, concave equation F ( D 2 u , t ) − u t = 0 F(D^{2}u,t)-u_{t}=0 in Q Q , where F F is measurable in t t .

Singular star-exponential functions

The ∗-exponential function e∗tieav∗u is defined in a transcendently extended Weyl algebra. In the Weyl ordering expression, this is given as the real analytic solution in t of ∂tFt(u,v)=ieav{(u+ℏia2)Ft(u+ℏia2,v)−ℏi2∂vFt(u+ℏia2,v)},F0(u,v)=1.. For generic initial functions, this equation can be solved for all t, but the uniqueness holds only for one direction.

Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary

The dual boundary element method in the real domain proposed by Hong and Chen in 1988 is extended to the complex variable dual boundary element method. This novel method can simplify the calculation for a hypersingular integral, and an exact integration for the influence coefficients is obtained. In addition, the Hadamard integral formula is obtained by taking the derivative of the Cauchy integral formula. The two equations (the Cauchy and Hadamard integral formula) constitute the basis for the complex variable dual boundary integral equations. After discretizing the two equations, the complex variable dual boundary element method is implemented. In determining the influence coefficients, the residue for a single-order pole in the Cauchy formula is extended to one of higher order in the Hadamard formula. In addition, the use of a simple solution and equilibrium condition is employed to check the influence matrices. To extract the finite part in the Hadamard formula, the extended residue theorem is employed. The role of the Hadamard integral formula is examined for the boundary value problems with a degenerate boundary. Finally, some numerical examples, including the potential flow with a sheet pile and the torsion problem for a cracked bar, are considered to verify the validity of the proposed formulation. The results are compared with those of real dual BEM and analytical solutions where available. A good agreement is obtained.

Propagation of analyticity of solutions to the Cauchy problem for Kirchhoff type equations

We shall give the local in time existence of the solutions in Gevrey classes to the Cauchy problem for Kirhhoff equations of p-laplacian type and investigate the propagation of analyticity of solutions for real analytic deta. When p=2, his equation as the global real analytic solution for the real analytic initial data.

Geometric Interpretation of the Phragmén–Lindelöf Estimates

In Section 3 we state a geometric version of Hörmander's criterion ( Invent. Math. 21(1973), 151–182) for global existence on open convex domains in R n of real analytic solutions to linear differential equations with constant coefficients. In Section 1 we discuss local hyperbolicity of analytic functions following the treatment given in Atiyah et al.( Acta Math. 124(1970), 109–189; 131(1973), 145–206). In Section 2 we give sufficient conditions for the Phragmén–Lindelöf implications to hold on the zero–sets of such functions. We also show that those conditions are necessary when each irreducible component of the zero–set has multiplicity ⩽2. The results on existence of real analytic solutions have already been stated in Zampieri ( J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31, No. 2 (1984), 372–390). The new contribution of the present paper consists in more accurate Phragmén– Lindelöf estimates and in a much shorter and more satisfactory proof. Also a new example is introduced; it explains in what sense our method is sharper than the others which have appeared in the literature (compare with Kawaı̈ ( J. Math. Soc. Japan 24(1972), 481–517)). This was the motivation for our research in this area.

The real-analytic solutions of the Abel functional equation

The real-analytic solutions of the Abel functional equation