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Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

Differential geometry. Global analysis --- Bifurcation theory. --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Addition. --- Algebraic equation. --- Analytic function. --- Analytic manifold. --- Atmospheric pressure. --- Banach space. --- Bernhard Riemann. --- Bifurcation diagram. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Boundedness. --- Canonical form. --- Cartesian coordinate system. --- Codimension. --- Compact operator. --- Complex analysis. --- Complex conjugate. --- Complex number. --- Connected space. --- Coordinate system. --- Corollary. --- Curvature. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Embedding. --- Equation. --- Euclidean division. --- Euler equations (fluid dynamics). --- Existential quantification. --- First principle. --- Fredholm operator. --- Froude number. --- Functional analysis. --- Hilbert space. --- Homeomorphism. --- Implicit function theorem. --- Integer. --- Linear algebra. --- Linear function. --- Linear independence. --- Linear map. --- Linear programming. --- Linear space (geometry). --- Linear subspace. --- Linearity. --- Linearization. --- Metric space. --- Morse theory. --- Multilinear form. --- N0. --- Natural number. --- Neumann series. --- Nonlinear functional analysis. --- Nonlinear system. --- Numerical analysis. --- Open mapping theorem (complex analysis). --- Operator (physics). --- Ordinary differential equation. --- Parameter. --- Parametrization. --- Partial differential equation. --- Permutation group. --- Permutation. --- Polynomial. --- Power series. --- Prime number. --- Proportionality (mathematics). --- Pseudo-differential operator. --- Puiseux series. --- Quantity. --- Real number. --- Resultant. --- Singularity theory. --- Skew-symmetric matrix. --- Smoothness. --- Solution set. --- Special case. --- Standard basis. --- Sturm–Liouville theory. --- Subset. --- Symmetric bilinear form. --- Symmetric group. --- Taylor series. --- Taylor's theorem. --- Theorem. --- Total derivative. --- Two-dimensional space. --- Union (set theory). --- Variable (mathematics). --- Vector space. --- Zero of a function.

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This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

Schrodinger equation. --- Wave mechanics. --- Equation, Schrödinger --- Schrödinger wave equation --- Electrodynamics --- Matrix mechanics --- Mechanics --- Molecular dynamics --- Quantum statistics --- Quantum theory --- Waves --- Differential equations, Partial --- Particles (Nuclear physics) --- Wave mechanics --- WKB approximation --- Schrödinger equation. --- Schrödinger, Équation de. --- Solitons. --- Abelian integral. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation. --- Asymptote. --- Asymptotic analysis. --- Asymptotic distribution. --- Asymptotic expansion. --- Banach algebra. --- Basis (linear algebra). --- Boundary (topology). --- Boundary value problem. --- Bounded operator. --- Calculation. --- Cauchy's integral formula. --- Cauchy's integral theorem. --- Cauchy's theorem (geometry). --- Cauchy–Riemann equations. --- Change of variables. --- Coefficient. --- Complex plane. --- Cramer's rule. --- Degeneracy (mathematics). --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differential equation. --- Differential operator. --- Dirac equation. --- Disjoint union. --- Divisor. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Energy minimization. --- Equation. --- Euler's formula. --- Euler–Lagrange equation. --- Existential quantification. --- Explicit formulae (L-function). --- Fourier transform. --- Fredholm theory. --- Function (mathematics). --- Gauge theory. --- Heteroclinic orbit. --- Hilbert transform. --- Identity matrix. --- Implicit function theorem. --- Implicit function. --- Infimum and supremum. --- Initial value problem. --- Integrable system. --- Integral curve. --- Integral equation. --- Inverse problem. --- Jacobian matrix and determinant. --- Kerr effect. --- Laurent series. --- Limit point. --- Line (geometry). --- Linear equation. --- Linear space (geometry). --- Logarithmic derivative. --- Lp space. --- Minor (linear algebra). --- Monotonic function. --- Neumann series. --- Normalization property (abstract rewriting). --- Numerical integration. --- Ordinary differential equation. --- Orthogonal polynomials. --- Parameter. --- Parametrix. --- Paraxial approximation. --- Parity (mathematics). --- Partial derivative. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Perturbation theory. --- Pole (complex analysis). --- Polynomial. --- Probability measure. --- Quadratic differential. --- Quadratic programming. --- Radon–Nikodym theorem. --- Reflection coefficient. --- Riemann surface. --- Simultaneous equations. --- Sobolev space. --- Soliton. --- Special case. --- Taylor series. --- Theorem. --- Theory. --- Trace (linear algebra). --- Upper half-plane. --- Variational method (quantum mechanics). --- Variational principle. --- WKB approximation. --- Schrödinger, Équation de.