In this work, we employ the $bar{partial}$-steepest descent method to investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with initial conditions in weighted Sobolev space $mathcal{H}(mathbb{R})$. The long time asymptotic behav
ior of the solution $q(x,t)$ is derived in a fixed space-time cone $S(y_{1},y_{2},v_{1},v_{2})={(y,t)inmathbb{R}^{2}: y=y_{0}+vt, ~y_{0}in[y_{1},y_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the soliton resolution conjecture of the WKI equation which includes the soliton term confirmed by $N(mathcal{I})$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$.
In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the $bar{partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long time asympto
tic behavior of the solution $q(x,t)$ in space-time soliton region. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ leading order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$. Our results also confirm the soliton resolution conjecture for the WKI equation with finite density initial data.
The Cauchy problem of the modified nonlinear Schr{o}dinger (mNLS) equation with the finite density type initial data is investigated via $overline{partial}$ steepest descent method. In the soliton region of space-time $x/tin(5,7)$, the long-time asym
ptotic behavior of the mNLS equation is derived for large times. Furthermore, for general initial data in a non-vanishing background, the soliton resolution conjecture for the mNLS equation is verified, which means that the asymptotic expansion of the solution can be characterized by finite number of soliton solutions as the time $t$ tends to infinity, and a residual error $mathcal {O}(t^{-3/4})$ is provided.
The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schr{o}dinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residue
s at the simple- and the second-poles. Further, the solution of RH problem is transformed into a closed system of algebraic equations, and the soliton solutions corresponding to the transmission coefficient $1/s_{11}(z)$ with an $N$-order pole are obtained by solving the algebraic system. Then, in a more general case, the transmission coefficient with multiple high-order poles is studied, and the corresponding solutions are obtained. In addition, for high-order pole, the propagation behavior of the soliton solution corresponding to a third-order pole is given as example.
We employ the $bar{partial}$-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb
{R})}$. The long time asymptotic behavior of the solution $u(x,t)$ is derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})={(x,t)inmathbb{R}^{2}: y=y_{0}+vt, ~y_{0}in[y_{1},y_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the CSP equation which includes the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$.
In this work, the $overline{partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(ma
thbb{R})}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=left{(x,t)in mathbb{R}timesmathbb{R}: x=x_{0}+vt ~text{with}~ x_{0}in[x_{1},x_{2}]right}$. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term $mathcal {O}(t^{-1/2})$ in the continuous spectrum, $mathcal {N}(mathcal {I})$ soliton solutions in the discrete spectrum and error order $mathcal {O}(t^{-3/4})$ from the $overline{partial}$ equation.
In this work, we employ the $bar{partial}$ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb{R})}$. The
large time asymptotic behavior of the solution $u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})={(x,t)inmathbb{R}^{2}: x=x_{0}+vt, ~x_{0}in[x_{1},x_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the cgNLS equations which contains the soliton term confirmed by $|mathcal{Z}(mathcal{I})|$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$.
The theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2mtimes2m$ $(mgeq 1)$ Lax pairs under the nonzero boundary conditions at infinity. In the direct pro
blem, by introducing a suitable uniform transformation we establish the proper complex $z$-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry of the equation. Moreover the asymptotic behavior of the Jost functions and scattering matrix needed in the inverse problem are analyzed via Wentzel-Kramers-Brillouin expansion. In the inverse problem, the generalized Riemann-Hilbert problem of the mmKdV equation is first established by using the analyticity of the modified eigenfunctions and scattering coefficients. The reconstruction formula of potential function with reflection-less case is derived by solving this Riemann-Hilbert problem and using the scattering data. In addition the dynamic behavior of the solutions for the focusing mmKdV equation including one- and two- soliton solutions are presented in detail under the the condition that the potential is scalar and the $2times2$ symmetric matrix. Finally, we provide some detailed proofs and weak version of trace formulas to show that the asymptotic phase of the potential and the scattering data.
In this work, we consider the generalized variable-coefficient nonlinear Schr{o}dinger equation with non-vanishing boundary conditions at infinity including the simple and double poles of the scattering coefficients. By introducing an appropriate Rie
mann surface and uniformization coordinate variable, we first convert the double-valued functions which occur in the process of direct scattering to single-value functions. Then, we establish the direct scattering problem via analyzing the analyticity, symmetries and asymptotic behaviors of Jost functions and scattering matrix derived from Lax pairs of the equation. Based on these results, a generalized Riemann-Hilbert problem is successfully established for the equation. The discrete spectrum and residual conditions, trace foumulae and theta conditions are investigated systematically including the simple poles case and double poles case. Moreover, the inverse scattering problem is solved via the Riemann-Hilbert approach. Finally, under the condition of reflection-less potentials, the soliton and breather solutions are well derived. Via evaluating the impact of each parameters, some interesting phenomena of these solutions are analyzed graphically.
We propose an heterogeneous multi-task learning framework for human pose estimation from monocular image with deep convolutional neural network. In particular, we simultaneously learn a pose-joint regressor and a sliding-window body-part detector in
a deep network architecture. We show that including the body-part detection task helps to regularize the network, directing it to converge to a good solution. We report competitive and state-of-art results on several data sets. We also empirically show that the learned neurons in the middle layer of our network are tuned to localized body parts.
