arXiv subjects

Jin-Jie Yang

Publications and source records attributed to Jin-Jie Yang.

At least 19 recordsLinked to original sources

arXiv2022

On the long-time asymptotic behavior of the Camassa-Holm equation in space-time solitonic regions

In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regions \begin{align*} m_t+2κq_x+3qq_x=2q_xq_{xx}+qq_{xx},~~m=q-q_{xx}+κ,\\ q(x,0)=q_0(x)\in H^{4,2}(\mathbb R),~~x\in\mathbb R, ~~t>0, \end{align*} where $κ$ is a positive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the CH equation with the initial boundary value condition. Furthermore, by developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, different long-time asymptotic expansions of the solution $q(x,t)$ are derived. Four asymptotic regions are divided in this work: For $ξ\in\left(-\infty,-\frac{1}{4}\right)\cup(2,\infty)$, the phase function $θ(z)$ has no stationary point on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by $N(j_0)$-soliton on discrete spectrum with residual error up to $O(t^{-1+2τ})$; For $ξ\in\left(-\frac{1}{4},0\right)$ and $ξ\in\left(0,2\right)$, the phase function $θ(z)$ has four and two stationary points on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by $N(j_0)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$. Our results also confirm the soliton resolution conjecture for the CH equation with weighted Sobolev initial data in space-time solitonic regions.

math.AP↗

arXiv2022

On the long-time asymptotic of the modified Camassa-Holm equation with nonzero boundary conditions in space-time solitonic regions

We investigate the long-time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with nonzero boundary conditions in different regions \begin{align*} &m_{t}+\left((u^2-u_x^2)m\right)_{x}=0,~~ m=u-u_{xx}, ~~ (x,t)\in\mathbb{R}\times\mathbb{R}^{+},\\ &u(x,0)=u_{0}(x),~~\lim_{x\to\pm\infty} u_{0}(x)=1,~~u_{0}(x)-1\in H^{4,1}(\mathbb{R}), \end{align*} where $m(x,t=0):=m_{0}(x)$ and $m_{0}(x)-1\in H^{2,1}(\mathbb{R})$. Through spectral analysis, the initial value problem of the mCH equation is transformed into a matrix RH problem on a new plane $(y,t)$, and then using the $\overline{\partial}$-nonlinear steepest descent method, we analyze the different asymptotic behaviors of the four regions divided by the interval of $ξ=y/t$ on plane $\{(y,t)|y\in(-\infty,+\infty), t>0\}$. There is no steady-state phase point corresponding to the regions $ξ\in(-\infty,-1/4)\cup(2,\infty)$. We prove that the solution of mCH equation is characterized by $N$-soliton solution and error on these two regions. In $ξ\in(-1/4,0)$ and $ξ\in(0,2)$, the phase function $θ(z)$ has eight and four steady-state phase points, respectively. We prove that the soliton resolution conjecture holds, that is, the solution of the mCH equation can be expressed as the soliton solution on the discrete spectrum, the leading term on the continuous spectrum, and the residual error. Our results also show that soliton solutions of the mCH equation with nonzero condition boundary are asymptotically stable.

math.AP↗

arXiv2022

Long-time asymptotic behavior of the nonlocal nonlinear Schrödinger equation with finite density type initial data

In this work, we employ the $\bar{\partial}$-steepest descent method to investigate the Cauchy problem of the nonlocal nonlinear Schrödinger (NNLS) equation with finite density type initial conditions in weighted Sobolev space $\mathcal{H}(\mathbb{R})$. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the NNLS equation with the finite density type initial boundary value condition. By developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution $q(x,t)$ in soliton region of space-time, $\left(\frac{x}{2t}\right)=ξ$ for any fixed $ξ=\in (1,K)$($K$ is a sufficiently large real constant), and give bounds for the error decaying as $|t|\rightarrow\infty$. Based on the resulting asymptotic behavior, the asymptotic approximation of the NNLS equation is characterized with the soliton term confirmed by $N(Λ)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.

nlin.SI↗

arXiv2021

Inverse scattering transform and dynamics of soliton solutions for nonlocal focusing modified Korteweg-de Vries equation

In this work, we mainly study the general $N$-soliton solutions of the nonlocal modified Korteweg-de Vries (mKdV) equation by utilizing the Riemann-Hilbert (RH) method. For the initial value belonging to Schwarz space, we firstly obtain the corresponding eigenfunctions and scattering data in the direct scattering process. Then we successfully establish a suitable RH problem of the nonlocal mKdV equation. The exact expression of the solution for the equation is derived via solving the RH problem. Using the symmetry of scattering data, the phenomena corresponding to different eigenvalues are analyzed, including bounded solutions, singular solutions, position solutions and kink solutions. Finally, the propagation path of the solution is observed, and the characteristic line is further used to analyze the continuity or other phenomena of the solution. The new dynamic behavior of the solution is observed by rotating the characteristic line at a certain angle.

nlin.SI↗

arXiv2021

Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data

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 behavior of the solution $q(x,t)$ is derived in a fixed space-time cone $S(y_{1},y_{2},v_{1},v_{2})=\{(y,t)\in\mathbb{R}^{2}: y=y_{0}+vt, ~y_{0}\in[y_{1},y_{2}], ~v\in[v_{1},v_{2}]\}$. 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 $O(t^{-\frac{3}{4}})$.

nlin.SI↗

arXiv2021

Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data

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 asymptotic 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.

math.AP↗

arXiv2021

Inverse scattering transform and multiple high-order pole solutions for the nonlocal focusing and defocusing modified Korteweg-de Vries equation with the nonzero boundary conditions

We extend the Riemann-Hilbert (RH) method to study the inverse scattering transformation and high-order pole solutions of the focusing and defocusing nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with nonzero boundary conditions (NZBCs) at infinity and successfully find its multiple soliton solutions with one high-order pole and multiple high-order poles. By introducing the generalized residue formula, we overcome the difficulty caused by calculating the residue conditions corresponding to the higher-order poles. In accordance with the Laurent series of reflection coefficient and oscillation term, the determinant formula of the high-order pole solution with NZBCs is established. Finally, combined with specific parameters, the dynamic propagation behaviors of the high-order pole solutions are further analyzed and some very interesting phenomena are obtained, including kink solution, anti kink solution, rational solution and breathing-soliton solution.

nlin.SI↗

arXiv2021

On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schrödinger equation

The Cauchy problem of the modified nonlinear Schrödinger (mNLS) equation with the finite density type initial data is investigated via $\overline{\partial}$ steepest descent method. In the soliton region of space-time $x/t\in(5,7)$, the long-time asymptotic 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.

math.AP↗

arXiv2021

Inverse scattering transform of the coupled modified Korteweg-de Vries equation with nonzero boundary conditions

In this work, we extend the Riemann-Hilbert (RH) method in order to study the coupled modified Korteweg-de Vries equation (cmKdV) under nonzero boundary conditions (NZBCs), and successfully find its solutions with their various dynamic propagation behaviors. In the process of spectral analysis, it is necessary to introduce Riemann surface to avoid the discussion of multi-valued functions, and to obtain the analytical and asymptotic properties needed to establish the RH problem. The eigenfunction have a column that is not analytic in a given region, so we introduce the auxiliary eigenfunction and the adjoint matrix, which is necessary to derive the analytical eigenfunctions. The eigenfunctions have three kinds of symmetry, which leads to three kinds of symmetry of the scattering matrix, and the discrete spectrum is also divided into three categories by us. The asymptoticity of the modified eigenfunction is derived. Based on the analysis, the RH problem with four jump matrices in a given area is established, and the relationship between the cmKdV equation and the solution of the RH problem is revealed. The residue condition of reflection coefficient with simple pole is established. According to the classification of discrete spectrum, we discuss the soliton solutions corresponding to three kinds of discrete spectrum classification and their propagation behaviors in detail.

nlin.SI↗

arXiv2021

Riemann-Hilbert problem for the nonlinear Schrödinger equation with multiple high-order poles under nonzero boundary conditions

The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schrödinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residues 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.

nlin.SI↗

arXiv2021

Soliton resolution for the complex short pulse equation with weighted Sobolev initial data

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 $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The long time asymptotic behavior of the solution $u(x,t)$ is derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})=\{(x,t)\in\mathbb{R}^{2}: y=y_{0}+vt, ~y_{0}\in[y_{1},y_{2}], ~v\in[v_{1},v_{2}]\}$. 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 $O(t^{-1})$.

math.AP↗

arXiv2021

Soliton resolution for the Hirota equation with weighted Sobolev initial data

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 $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=\left\{(x,t)\in \mathbb{R}\times\mathbb{R}: x=x_{0}+vt ~\text{with}~ x_{0}\in[x_{1},x_{2}]\right\}$. 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.

math.AP↗

arXiv2020

Soliton resolution for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data

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 $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The large time asymptotic behavior of the solution $u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})=\{(x,t)\in\mathbb{R}^{2}: x=x_{0}+vt, ~x_{0}\in[x_{1},x_{2}], ~v\in[v_{1},v_{2}]\}$. 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 $O(t^{-\frac{3}{4}})$.

math.AP↗

arXiv2020

Riemann-Hilbert problem for the sextic nonlinear Schrödinger equation with non-zero boundary conditions

We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schrödinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization coordinate variable in order to avoid multi-value problems. Based on a new complex plane, the direct scattering problem perform a detailed analysis of the analytical, asymptotic and symmetry properties of the Jost functions and the scattering matrix. Then, a generalized Riemann-Hilbert problem (RHP) is successfully established from the results of the direct scattering transform. In the inverse scattering problem, we discuss the discrete spectrum, residue condition, trace formula and theta condition under simple poles and double poles respectively, and further solve the solution of a generalized RHP. Finally, we derive the solution of the equation for the cases of different poles without reflection potential. In addition, we analyze the localized structures and dynamic behaviors of the resulting soliton solutions by taking some appropriate values of the parameters appeared in the solutions.

nlin.SI↗

arXiv2020

Inverse scattering transform and multi-solition solutions for the sextic nonlinear Schrödinger equation

In this work, we consider the inverse scattering transform and multi-solition solutions of the sextic nonlinear Schrödinger equation. The Jost functions of spectrum problem are derived directly, and the scattering data with $t=0$ are obtained according to analyze the symmetry and other related properties of the Jost functions. Then we take use of translation transformation to get the relation between potential and kernel, and recover potential according to Gel'fand-Levitan-Marchenko (GLM) integral equations. Furthermore, the time evolution of scattering data is considered, on the basic of that, the multi-solition solutions are derived. In addition, some solutions of the equation are analyzed and revealed its dynamic behavior via graphical analysis, which could be enriched the nonlinear phenomena of the sextic nonlinear Schrödinger equation.

nlin.SI↗

arXiv2020

Inverse scattering transform and soliton solutions for the modified matrix Korteweg-de Vries equation with nonzero boundary conditions

The theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2m\times2m$ $(m\geq 1)$ Lax pairs under the nonzero boundary conditions at infinity. In the direct problem, 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 $2\times2$ 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.

nlin.SI↗

arXiv2020

Riemann-Hilbert approach to the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions

In this work, we consider the generalized variable-coefficient nonlinear Schrödinger equation with non-vanishing boundary conditions at infinity including the simple and double poles of the scattering coefficients. By introducing an appropriate Riemann 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.

nlin.SI↗

arXiv2020

Riemann-Hilbert approach to the inhomogeneous fifth-order nonlinear Schrödinger equation with non-vanishing boundary conditions

We consider the inhomogeneous fifth-order nonlinear Schrödinger (ifoNLS) equation with nonzero boundary condition in detailed. Firstly, the spectral analysis of the scattering problem is carried out. A Riemann surface and affine parameters are first introduced to transform the original spectral parameter to a new parameter in order to avoid the multi-valued problem. Based on Lax pair of the ifoNLS equation, the Jost functions are obtained, and their analytical, asymptotic, symmetric properties, as well as the corresponding properties of the scattering matrix are established systematically. For the inverse scattering problem, we discuss the cases that the scattering coefficients have simple zeros and double zeros, respectively, and we further derive their corresponding exact solutions. Moreover, some interesting phenomena are found when we choose some appropriate parameters for these exact solutions, which is helpful to study the propagation behavior of these solutions.

nlin.SI↗