Testing the Dimensionality of Policy Shocks

This paper provides a nonparametric test for deciding the dimensionality of a policy shock as manifest in the abnormal change in asset returns' stochastic covariance matrix, following the release of a macroeconomic announcement. We use high-frequency data in local windows before and after the event to estimate the covariance jump matrix, and then test its rank. We find a one-factor structure in the covariance jump matrix of the yield curve resulting from the Federal Reserve's monetary policy shocks prior to the 2007-2009 financial crisis. The dimensionality of policy shocks increased afterwards due to the use of unconventional monetary policy tools.

Dynamic Behaviors of Soliton Solutions for a Three-Coupled Lakshmanan-Porsezian-Daniel Model

In this paper, we use the Riemann-Hilbert (RH) approach to examine the integrable three-coupled Lakshmanan-Porsezian-Daniel (LPD) model, which describe the dynamics of alpha helical protein with the interspine coupling at the fourth-order dispersion term. Through the spectral analysis of Lax pair, we construct the higher order matrix RH problem for the three-coupled LPD model, when the jump matrix of this particular RH problem is a 4×4 unit matrix, the exact N-soliton solutions of the three-coupled LPD model can be exhibited. As special examples, we also investigate the nonlinear dynamical behaviors of the single-soliton, two-soliton, three-soliton and breather soliton solutions. Finally, an integrable generalized N-component LPD model with its linear spectral problem is discussed.

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.

N-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert approach

We mainly study [Formula: see text]-soliton solutions for the Maxwell–Bloch equations via the Riemann–Hilbert (RH) approach in this paper. The relevant RH problem has been constructed by performing spectral analysis of Lax pair. Then the jump matrix of the Maxwell–Bloch equations has been obtained. Finally, we gain the exact solutions of the Maxwell–Bloch equations by solving the special RH problem with reflectionless case.

Multi-Soliton Solutions of the N-Component Nonlinear Schrodinger Equations via Riemann-Hilbert Approach*

In this paper, we utilize the Riemann-Hilbert approach to discuss multi-soliton solutions of the N-component nonlinear Schrodinger equations. Firstly, by transformed Lax pair, we construct the matrix valued functions P1,2 that satisfy the analyticity and normalization and the corresponding jump matrix can be determined. Then, in the reflectionless case, we get the multi-soliton solutions ql(l = 1, ..., N) of the N-component nonlinear Schrodinger equations, which are related to spectral parameters. Particularly, the 2-soliton solutions q1, q2 and q3 of the three-component nonlinear Schrodinger equations are given and the corresponding 2-soliton diagrams are drawn.

Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies

This paper aims to explore nonlocal complex reverse-spacetime modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems and to construct their soliton solutions by the inverse scattering transforms. The corresponding inverse scattering problems are formulated by building the associated Riemann-Hilbert problems. A formulation of solutions to specific Riemann-Hilbert problems, with the jump matrix being the identity matrix, is established, where eigenvalues could equal adjoint eigenvalues, and thus N-soliton solutions to the nonlocal complex reverse-spacetime mKdV hierarchies are obtained from the reflectionless transforms.

Riemann–Hilbert problems of a nonlocal reverse-time six-component AKNS system of fourth order and its exact soliton solutions

We investigate the solvability of an integrable nonlinear nonlocal reverse-time six-component fourth-order AKNS system generated from a reduced coupled AKNS hierarchy under a reverse-time reduction. Riemann–Hilbert problems will be formulated by using the associated matrix spectral problems, and exact soliton solutions will be derived from the reflectionless case corresponding to an identity jump matrix.

A Riemann-Hilbert Approach to the Multicomponent Kaup-Newell Equation

A Riemann-Hilbert approach is developed to the multicomponent Kaup-Newell equation. The formula is presented of N -soliton solutions through an identity jump matrix related to the inverse scattering problems with reflectionless potential.

N-Soliton Solutions for the NLS-Like Equation and Perturbation Theory Based on the Riemann–Hilbert Problem

In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann–Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann–Hilbert (R–H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of det P + and det P − ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R–H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R–H problem, the evolution equation of the R–H data with the perturbation is derived.

N-Soliton Solutions of the Coupled Kundu Equations Based on the Riemann-Hilbert Method

The Kundu equation, which can be used to describe many phenomena in physics and mechanics, has crucial theoretical meaning and research value. In previous studies, the single Kundu equation has been investigated by the Riemann-Hilbert method, but few researchers have focused on the coupled Kundu equations. To our knowledge, many phenomena in nature can be only described by coupled equations, such as species competition and signal interactions. In this paper, we discuss N-soliton solutions of the coupled Kundu equations according to the Riemann-Hilbert method. Starting from the spectral problem, the coupled Kundu equations are generated, and the Riemann-Hilbert problem is presented. When the jump matrix of the Riemann-Hilbert problem is the identity matrix, the N-soliton solutions of the coupled Kundu equations can be expressed explicitly.