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This book presents the construction of an asymptotic technique for solving the Liouville equation, which is to some degree an analogue of the Enskog-Chapman technique for solving the Boltzmann equation. Because the assumption of molecular chaos has been given up at the outset, the macroscopic variables at a point, defined as arithmetic means of the corresponding microscopic variables inside a small neighborhood of the point, are random in general. They are the best candidates for the macroscopic variables for turbulent flows. The outcome of the asymptotic technique for the Liouville equation reveals some new terms showing the intricate interactions between the velocities and the internal energies of the turbulent fluid flows, which have been lost in the classical theory of BBGKY hierarchy.

Statistical mechanics. --- Sturm-Liouville equation. --- Mechanics --- Mechanics, Analytic --- Quantum statistics --- Statistical physics --- Thermodynamics --- Liouville-Sturm equation --- Boundary value problems --- Differential equations

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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.