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We consider soliton gas solutions of the modified Korteweg-de Vries (mKdV) equation, where the point spectrum of the condensate is located within a bounded domain in the upper half-plane. We first demonstrate that when the domain is a quadrature and the soliton density is an analytic function, the corresponding deterministic soliton gas coincides with a finite number of solitons, which we call this effect soliton shielding. When the domain is an ellipse and the soliton density is analytic, the corresponding deterministic soliton gas reduces the spectral data to the segment joining the foci. The initial datum of this Cauchy problem is asymptotically step-like oscillatory, described by a periodic elliptic function as \( x \to +\infty \), and it vanishes exponentially fast as \( x \to -\infty \)., Comment: 24 pages

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system and admits a $3\times3$ matrix Lax pair. In this manuscript, we study the soliton resolution and large time behavior of solutions to the Cauchy problem of the DP equation with generic initial data in Schwarz space. Employing the $\bar \partial$-generalization of the Deift-Zhou nonlinear steepest descent method, we deduce different long time asymptotic expansions of the solution $u(x,t)$ in two distinct types of space-time regions. This result verifies the soliton resolution conjecture and asymptotic stability of $N$-soliton solutions for the DP equation., Comment: 40 pages

We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial data \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber \\ &u(x,0)=u_0(x)\to \left\{ \begin{array}{ll} A_1, &\ x\to+\infty,\\[5pt] A_2, &\ x\to-\infty, \end{array}\right.\nonumber \end{align} where $A_1$ and $A_2$ are two positive constants. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the step-like initial problem is characterized via the solution of a RH problem in the new scale $(y,t)$. We adopt double coordinates $(\xi, c)$ to divide the half-plane $\{ (\xi,c): \xi \in \mathbb{R}, \ c> 0, \ \xi=y/t\}$ into four asymptotic regions. Further using the Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(y,t)$ in different space-time regions by the different choice of g-function. The corresponding leading asymptotic approximations are given with the slow/fast decay step-like background wave in genus-0 regions and elliptic waves in genus-2 regions. The second term of the asymptotics is characterized by Airy function or parabolic cylinder model. Their residual error order is $\mathcal{O}(t^{-1})$ or $\mathcal{O}(t^{-2})$ respectively., Comment: 86 pages

We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with a nonzero background $$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1. \end{align}$$ Recently, for the space-time region $|x/(2t)|<1$ which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form \begin{align} q(x,t)= T(\infty)^{-2} q^{sol,N}(x,t) + \mathcal{O}(t^{-1 }),\label{res1} \end{align} whose leading term is N-soliton and the second term $\mathcal{O}(t^{-1})$ is a residual error from a $\overline\partial$-equation. In this paper, we are interested in the large-time asymptotics in the space-time region $ |x/(2t)|>1$ which is outside the soliton region, but there will be two stationary points appearing on the jump contour $\mathbb{R}$. We found a asymptotic expansion that is different from (\ref{res1}) $$\begin{align} q(x,t)= e^{-i\alpha(\infty)} \left(1 +t^{-1/2} h(x,t) \right)+\mathcal{O}\left(t^{-3/4}\right),\label{res2} \end{align}$$ whose leading term is a nonzero background, the second $t^{-1/2}$ order term is from continuous spectrum and the third term $\mathcal{O}(t^{-3/4})$ is a residual error from a $\overline\partial$-equation.The above two asymptotic results (\ref{res1}) and (\ref{res2}) imply that the region $ |x/(2t)|<1$ considered by Cuccagna and Jenkins is a fast decaying soliton solution region, while the region $ |x/(2t)|>1$ considered by us is a slow decaying nonzero background region., Comment: 53 pages

In this paper, we use the $\bar{\partial}$ steepest descent method to study the initial value problem for focusing nonlinear Schr\"odinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we shall characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; further we formulate an appropriate enlarged RH problem; after a series of deformations, the RH problem is transformed into a solvable model. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone: %as $t \to \infty$, \begin{equation*} \mathcal{S}(x_1,x_2,v_1,v_2):=\left\lbrace (x,t)\in \mathbb{R}^2: x=x_0+vt, \ x_0\in[x_1,x_2]\text{, }v\in[v_1,v_2]\right\rbrace. \end{equation*} Observing the result indicates that the solution of fNLS equation in this case satisfies the soliton resolution conjecture. The leading order term of this solution includes a high-order pole-soliton whose parameters are affected by soliton-soliton interactions through the cone and soliton-radiation interactions on continuous spectrum. The error term of this result is up to $\mathcal{O}(t^{-3/4})$ which comes from the corresponding $\bar{\partial}$ equation., Comment: 44 pages

In this paper, we study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation in the solitonic region \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}+\kappa u_{x}=0, \quad m=u-u_{x x}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align} where $\kappa$ is a positive constant. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a Riemann-Hilbert (RH) problem. Further using the $\overline\partial$ generalization of Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(x,t)$ in different space-time solitonic region of $x/t$. These asymptotic approximations can be characterized with an $N(\Lambda)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region with diverse residual error order from $\overline\partial$ equation: $\mathcal{O}(|t|^{-1+2\rho})$ for $\xi=\frac{y}{t}\in(-\infty,-0.25)\cup(2,+\infty)$ and $\mathcal{O}(|t|^{-3/4})$ for $\xi=\frac{y}{t}\in(-0.25,2)$. Our results also confirm the soliton resolution conjecture and asymptotically stability of N-soliton solutions for the mCH equation., Comment: 90 pages. arXiv admin note: substantial text overlap with arXiv:2012.15496

In this paper, we apply $\overline\partial$ steepest descent method to study the Cauchy problem for the derivative nonlinear Schr\"odinger equation with nonzero boundary conditions \begin{align} &iq_{t}+q_{xx}+i\sigma(|q|^2q)_{x}=0,\\ & (x,0) = q_0(x), \quad\lim_{x\to\pm\infty} q_0(x) = q_\pm,\end{align} where $|q_\pm|=1$. Based on the spectral analysis of the Lax pair, we express the solution of the derivative nonlinear Schr\"odinger equation in terms of solutions of a Riemann-Hilbert problem.In a fixed space-time solitonic region $-3

We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schr\"{o}dinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation, Comment: 35 pages

We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation $$ u_{xt}+\alpha\beta^2u-2i\alpha\beta u_x-\alpha u_{xx}-i\alpha\beta^2|u|^2u_x=0 \label{cs} $$ with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at $k=0$. (ii) four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone $$\mathcal{C}\left(x_{1}, x_{2}, v_{1}, v_{2}\right)=\left\{(x, t) \in \mathbb{R}^{2} | x=x_{0}+v t, x_{0} \in\left[x_{1}, x_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\},$$ the long-time asymptotic behavior of the solution $u(x,t)$ for the focusing FL equation can be characterized with an $N(\mathcal{I})$-soliton on discrete spectrums and a leading order term $\mathcal{O}(|t|^{-1/2})$ on continuous spectrum up to a residual error order $\mathcal{O}(|t|^{-3/4})$. The main tool is the $\overline{\partial}$ nonlinear steepest descent method and the $\overline{\partial}$-analysis., Comment: 71 pages

In this paper, we study the Cauchy problem for the focusing nonlinear short-pluse equation by using $\overline\partial$ steepest descent method. \begin{align} &u_{xt}=u+\frac{1}{6}(u^3)_{xx}, \nonumber\\ &u(x,0)=u_0(x)\in H^{1,1}(R),\nonumber \end{align} where $H^{1,1}(R)$ is a weighted Sobolev space. Because the spectral variable z is the same order in the WKI-type Lax pair, we construct the solution of SP equation in the new scale $(y,t)$, whereas the original scale $(x,t)$ is given in terms of functions in the new scale and the solution of Riemann-Hilbert problem. In any fixed space-time cone of the new scale $(y,t)$ which stratify that $v_1\leq v_1 \in R^-$ and $\xi=\frac{y}{t}<0$, \begin{equation} C(y_1,y_2,v_1,v_2) = \left\lbrace (y,t) \in R^2|y=y_0+vt, y_0 \in[y_1,y_2]\text{, } v\in[v_1,v_2]\right\rbrace, \nonumber \end{equation} we compute the long time asymptotic expansion of the solution $u(x,t)$, which prove soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an $N(I)$-soliton whose parameters are modulated by a sum of localied soliton-soliton interactions as one moves through the cone; the second $t^{-1/2}$ order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order $\mathcal{O}(|t|^{-1})$ from a $\overline\partial$ equation. Our results also show that soliton solutions of short-pluse equation are asymptotically stable., Comment: 48 pages. arXiv admin note: text overlap with arXiv:1912.10358

In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^\alpha(1-x)^\beta, $$ where $t \ge 0$, $\alpha >0$, $\beta >0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, different asymptotic behaviors at the hard edge $0$ and $1$ were found with different scaling schemes. Specifically, the uniform asymptotic behavior can be expressed as a Airy function in the neighborhood of point $1$ as $\zeta= 2n^2t \to \infty, n\to \infty$, while it is given by a Bessel function as $\zeta \to 0, n \to \infty$. { II. Universality:} We respectively calculate the limit of the eigenvalue correlation kernel in the bulk of the spectrum and at the both side of hard edge, which will involve a $\psi$-functions associated with a particular Painlev$\acute{e}$ \uppercase\expandafter{\romannumeral3} equation near $x=\pm 1$. Further, we also prove the $\psi$-funcation can be approximated by a Bessel kernel as $\zeta \to 0$ compared with a Airy kernel as $\zeta \to \infty$. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problems., Comment: 48 pages

In this article, we focus on the inverse scattering transformation for the Fokas-Lenells (FL) equation with nonzero boundary conditions via the Riemann-Hilbert (RH) approach. Based on the Lax pair of the FL equation, the analyticity and symmetry, asymptotic behavior of Jost solutions and scattering matrix are discussed in detail. With these results, we further present a generalized RH problem, from which a reconstruction formula between the solution of the FL equation and the Riemann-Hilbert problem is obtained. The N-soliton solutions of the FL equation is obtained via solving the RH problem., Comment: 21 pages

In this paper, we consider the Cauchy problem for the modified NLS equation. Using nonlinear steepest descent method and combining the Dbar-analysis, we show that inside any fixed cone, the long time asymptotic behavior of the solution for the modified NLS equation can be characterized with an soliton on discrete spectrum and leading order aasymptotic term on continuous spectrum up to an residual error order O(t^{-3/4})., Comment: 56 pages

We study the Cauchy problem for the focusing nonlinear Kundu-Eckhaus equation and construct long time asymptotic expansion of its solution in fixed space-time cone with $C(x_1,x_2,v_1,v_2)=\{(x,t)\in\Re^2:x=x_0+vt$ $x_0\in[x_1,x_2],v\in[v_1,v_2] \}$. By using the inverse scattering transform, Riemann-Hilbert approach and $\overline\partial$ steepest descent method we obtain the lone time asymptotic behavior of the solution, at the same time we obtain the solitons in the cone compare with the all N-soliton the residual error up to order $\mathcal{O}(t^{-3/4})$., Comment: 33 pages. arXiv admin note: text overlap with arXiv:1604.07436 by other authors

In this paper, we investigate the long-time asymptotic behavior of the solution to the initial value problem for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. The equation is known to be integrable, which we mean it admits an Lax pair. We formulate the initial value problem as an associate vector Riemann-Hilbert problem, which allows us to give a parametric representation of the solution to the initial value problem in terms of the solution of the Riemann-Hilbert problem. And then by adopting the nonlinear steepest descent method, we can get the explicit leading order asymptotic of the solution as time goes to infinity., Comment: This result was reported in the forth China-Japan Joint Workshop on Integrable Systems held on August 19-22, 2019

We concentrate on inverse scattering transformation for the Sasa-Satsuma equation with $3\times 3$ matrix spectral and nonzero boundary condition in this article. To circumvent multi valuedness of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter $k$ into a single-valued parameter $z$. The analyticity of the Jost eigenfunctions and scattering coefficients of Lax pair for the SS equation are analyzed in details. According to the analyticity of eigenfunctions and scattering coefficients, the $z$-complex plane is divided into four analytic regions $D_j, \ j=1, 2, 3, 4$. Since the second column of Jost eigenfunctions is analytic in $D_{j}, \ j=1, 2, 3, 4$, but in upper-half or lower-half plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in $D_{j}$. We find that for the eigenfunctions, scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries, which characterize the distribution of discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kind jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from which $N$-soliton solutions is obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As application of the $N$-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum., Comment: 50 pages

In this article, we focus on investigating the focusing Kundu-Eckhaus equation with nonzero boundary condition. A appropriate two-sheeted Riemann surface is introduced to map the spectral parameter $k$ into a single-valued parameter $z$. Starting from the Lax pair of Kundu-Eckhaus equation,two kind of Jost solutions are construed. Further their asymptotic, analyticity, symmetries as well as spectral matrix are detailed analyzed. It is shown that the solution of Kundu-Eckhaus equation with nonzero boundary condition can characterized with a matrix Riemann-Hilbert problem. Then a formula of $N$-soliton solutions is derived by solving Riemann-Hilbert problem. As applications, the first-order explicit soliton solution is obtained., Comment: 26 pages

The modified nonlinear Schr\"{o}dinger (NLS) equation was proposed to describe the nonlinear propagation of the Alfven waves and the femtosecond optical pulses in a nonlinear single-mode optical fiber. In this paper, the inverse scattering transform for the modified NLS equation with non-vanishing asymptotic boundary at infinity is presented. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter $k$ into a single-valued parameter $z$. The asymptotic behaviors, analyticity and the symmetries of the Jost solutions of Lax pair for the modified NLS equation, as well as the spectral matrix are analyzed in details. Then a matrix Riemann-Hilbert (RH) problem associated with the problem of nonzero asymptotic boundary conditions is established, from which $N$-soliton solutions is obtained via the corresponding reconstruction formulae. As an illustrate examples of $N$-soliton formula, two kinds of one-soliton solutions and three kinds of two-soliton solutions are explicitly presented according to different distribution of the spectrum. The dynamical feature of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions. Finally, we analyze the differences between our results and those on zero boundary case., Comment: 38 pages

In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation (nonlocal mKdV) \[q_t(x,t)+q_{xxx}(x,t)-6q(x,t)q(-x,-t)q_x(x,t)=0,\] which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal mKdV equation. In contrast with the classical mKdV equation, we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results., Comment: 27 pages

In this paper, we study Hankel determinants generated from a perturbed Laguerre weight function, Under the double scaling scheme, we give the uniform asymptotic approximations of Hankel determinants in terms of a solution of a third-order nonlinear differential equation, which is equivalent to a particular Painleve V equation. In fact, this Painleve V equation is equivalent to the general Painleve III equation. The asymptotic approximations of the leading coefficients and the recurrence coefficients for the corresponding orthogonal polynomials also involve the painleve V equation. The asymptotic analysis is based on our earlier results by using the Deift-Zhou nonlinear steepest descent method., Comment: 20 pages

In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of $$w(x,t)=(x+t)^{\lambda}x^{\alpha}e^{-x}$$ with $ x\geq 0,\ t>0,\ \alpha>0,\ \alpha+\lambda+1 > 0.$ The Deift-Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling $s=4nt,$ $n\to \infty,$ $t\to 0 $ such that $s$ is positive and finite, at the hard edge, the limiting kernel can be described by the $\varphi$-function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlev\'e V (shorted as P$_{\rm V}$) transcendent via a simple transformation. Moreover, this P$_{\rm V}$ transcendent is equivalent to a general Painlev\'e P$_{\rm III}$ transcendent. For large $s,$ the P$_{\rm V}$ kernel reduces to the Bessel kernel $\mathbf{J}_{\alpha+\lambda}.$ For small $s,$ the P$_{\rm V}$ kernel reduces to another Bessel kernel $\mathbf{J}_\alpha.$ At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight., Comment: 53 pages

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with $3 \times 3$ Lax pairs. The solution can be expressed in terms of the solution of a $3\times3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions $s(\lambda)$, $S(\lambda)$ and $S_L(\lambda)$, which arising from the initial values at $t=0$, boundary values at $x=0$ and boundary values at $x=L$, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. The relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval tends to infinity., Comment: arXiv admin note: substantial text overlap with arXiv:1509.02617, arXiv:1304.4586; text overlap with arXiv:1108.2875 by other authors

We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the Sasa-Satsuma(SS) equation on the finite interval. Assume that the solution existes, we show that this solution can be expressed in terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$, $S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. Here, we analyze the global relation to characterize the unknown boundary values in terms of the given initial and boundary data., Comment: arXiv admin note: substantial text overlap with arXiv:1509.02617, arXiv:1304.4586; text overlap with arXiv:1108.2875 by other authors

In a previous work, we show that the solution of the initial-boundary value problem for the two-component nonlinear Schr\"odinger equation on the finite interval can be expressed in terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$, $S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. Here, we use a Gelfand-Levitan-Marchenko representation to derive an expression for the generalized Dirichlet-to-Neumann map to characterize the unknown boundary values in physical domain, which is different from the approach, in fact it analyzed the global relation in spectral domain, used in the previous work. And, we can show that these two representations are equivalent., Comment: arXiv admin note: text overlap with arXiv:1109.4937, arXiv:1008.5379 by other authors

We present an approach for analyzing initial-boundary value problems which is formulated on the finite interval ($0\le x\le L$, where $L$ is a positive constant) for integrable equations whose Lax pairs involve $3\times 3$ matrices. Boundary value problems for integrable nonlinear evolution PDEs can be analyzed by the unified method introduced by Fokas and developed by him and his collaborators. In this paper, we show that the solution can be expressed in terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$,$S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, respectively. However, these spectral functions are not independent, they satisfy a global relation. Here, we show that the characterization of the unknown boundary values in terms of the given initial and boundary data is explicitly described for a nonlinear evolution PDE defined on the interval. Also, we show that in the limit when the length of the interval tends to infity, the relevant formulas reduce to the analogous formulas obtained for the case of boundary value problems formulated on the half-line., Comment: arXiv admin note: substantial text overlap with arXiv:1304.4586; text overlap with arXiv:1108.2875 by other authors

In this paper, we analyze the large n-limit for random matrix with external source with three distinct eigenvalues. And we confine ourselves in the Hermite case and the three distinct eigenvalues are $-a,0,a$. For the case $a^2>3$, we establish the universal behavior of local eigenvalue correlations in the limit $n\rightarrow \infty$, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. The result can be obtained by analyzing $4\times 4$ Riemann-Hilbert problem via nonlinear steepest decent method., Comment: 41 pages. arXiv admin note: substantial text overlap with arXiv:math-ph/0402042 by other authors

We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line. This equation can be viewed as the short wave model for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can be recovered from its initial and boundary values via the solution of a 3\times 3 vector Riemann-Hilbert problem formulated in the complex plane of a spectral parameter z., Comment: 25 pages. arXiv admin note: text overlap with arXiv:1204.5252, arXiv:1311.0495 by other authors

We extend Gesztesy-Holden's method to 2+1 dimensional case to obtain a unified construction to the algebro-geometric solutions of the whole modified Kadomtsev-Petviashvili (mKP) hierarchy. Our tools include the relations between solutions of the Gerdjikov-Ivanov (GI) and mKP hierarchy, the Baker-Akhiezer functions in 2+1 dimensions, a special function $\psi_1(P)\psi_2(P^*)$ on $X\times \mathbb{R}^3$ and Dubrovin-type equations for auxiliary divisors., Comment: 42 pages. arXiv admin note: substantial text overlap with arXiv:1410.7516

Based on the idea of symmetric constraint, we apply the Gesztesy-Holden's method to derive explicit representations of the Baker-Ahkiezer function $\psi_1$ of the KP hierarchy, from which we provide theta function representations of algebro-geometric solutions for the whole Kadomtsev-Petviashvili (KP) hierarchy. This provides a approach to obtain some special subclasses of algebro-geometric solutions for the KP hierarchy and other high dimensional hierarchy of equations., Comment: 44 pages

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through studying an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS2 hierarchy., Comment: 46 pages. accepted for publication J Nonl Math Phys, 2014. arXiv admin note: substantial text overlap with arXiv:1406.6153, arXiv:1207.0574, arXiv:1205.6062; and with arXiv:nlin/0105021 by other authors

In this paper, we consider the Harry-Dym equation on the line with decaying initial value. The Fokas unified method is used to construct the solution of the Harry-Dym equation via a $2 \times 2$ matrix Riemann Hilbert problem in the complex plane. Further, one-cups soltion solution is expressed in terms of solutions of the Riemann Hilbert problem., Comment: 17 pages

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Camassa-Holm Dym (CHD2) hierarchy. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire CHD2 hierarchy., Comment: 39 pages. arXiv admin note: substantial text overlap with arXiv:1207.0574, arXiv:1205.6062, arXiv:1305.0122; and with arXiv:nlin/0105021 by other authors

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x,y,z. In this paper, the Clarkson-Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found., Comment: Key Words: Incompressible Euler equations, the Clarkson-Kruskal method, similarity reductions, nonlinear exact solutions

In a previous study, we obtained the algebro-geometric solutions and $n$-dark solitons of Forkas-Lenells (FL) hierarchy using algebro-geometric method. In this paper, we construct physically relevant classes of solutions for FL hierarchy by studying the reality conditions for $q=\pm \bar{r}$ based on the idea of Vinikov's homological basis., Comment: 27 pages

We use the Deift-Zhou method to obtain, in the solitonless sector, the leading order asymptotic of the solution to the Cauchy problem of the Fokas-Lenells equation as $t\ra+\infty$ on the full-line., Comment: 47 pages. arXiv admin note: substantial text overlap with arXiv:solv-int/9701001 by other authors

This paper is dedicated to provide theta function representations of algebro-geometric solutions for the Fokas-Lenells (FL) hierarchy through studying an algebro-geometric initial value problem. Further, we reduce these solutions into $N$-dark solutions through the degeneration of associated Riemann surfaces., Comment: 48 pages. arXiv admin note: text overlap with arXiv:nlin/0702058, arXiv:nlin/0105021, arXiv:nlin/0611055 by other authors

We consider the Cauchy problem for the Gerdjikov-Ivanov(GI) type of the derivative nonlinear Schr\"odinger (DNLS) equation: $$iq_t+q_{xx}-iq^2\bar{q}_x+\frac{1}{2}|q|^4{q}=0.$$ with steplike initial data: $q(x,0)=0$ for $x\le 0$ and $q(x,0)=Ae^{-2iBx}$ for $x>0$,where $A>0$ and $B\in \R$ are constants.The paper aims at studying the long-time asymptotics of the solution to this problem.We show that there are four regions in the half-plane $-\infty0$,where the asymptotics has qualitatively different forms:a slowly decaying self-similar wave of Zakharov-Manakov type for $x>-4tB$, a plane wave region:$x<-4t(B+\sqrt{2A^2(B+\frac{A^2}{4})})$, an elliptic region:$-4t(B+\sqrt{2A^2(B+\frac{A^2}{4})})

We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the Sasa-Satsuma(SS) equation: $iq_T+\frac{1}{2}q_{XX}+|q|^2q+i\eps (q_{XXX}+6|q|^2q_X+3q(|q|^2)_X)=0$ on the half-line. And we also analysis the global relation in this paper., Comment: 33 pages. arXiv admin note: substantial text overlap with arXiv:1304.4339; and with arXiv:1108.2875 by other authors

We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the three-wave equation: \[p_{ij,t}-\frac{b_i-b_j}{a_i-a_j}p_{ij,x}+\sum_k(\frac{b_k-b_j}{a_k-a_j}-\frac{b_i-b_k}{a_i-a_k})p_{ik}p_{kj}=0,\quad i,j,k=1,2,3.\] on the half-line., Comment: arXiv admin note: substantial text overlap with arXiv:1108.2875 by other authors

We present a new Riemann-Hilbert problem formalism for the initial value problem for the derivative nonlinear Schr\"odinger (DNLS) equation on the line. We show that the solution of this initial value problem can be obtained from the solution of some associated Riemann-Hilbert problem. This new Riemann-Hilbert problem for the DNLS equation will lead us to use nonlinear steepest-descent/stationary phase method or Deift-Zhou method to derive the long-time asymptotic for the DNLS equation on the line., Comment: 41 pages

We provide a detailed treatment of relativistic Lotka-Volterra hierarchy and a kind of initial value problem with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve hyperelliptic curve $\mathcal{K}_n$ associated with the Burchnall-Chaundy polynomial, Dubrovin-type equations for auxiliary divisors and associated trace formulas. With the help of a foundamental meromorphic function $\tilde{\phi}$ on $\mathcal{K}_p$ and trace formulas, the complex-valued algebro-geometric solutions of of RLV hierarchy are derived., Comment: 42 pages. arXiv admin note: substantial text overlap with arXiv:1204.4240; and with arXiv:nlin/0702058 by other authors

This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the Hunter-Saxton (HS) hierarchy through studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the HS hierarchy, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire HS hierarchy., Comment: 50 pages. arXiv admin note: substantial text overlap with arXiv:1205.6062

This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the modified Camassa-Holm (MCH) hierarchy through %and studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the MCH hierarchy, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire MCH hierarchy., Comment: 59 pages. arXiv admin note: text overlap with arXiv:solv-int/9809004, arXiv:solv-int/9707010, arXiv:solv-int/9707009

We use the Fokas method to analyze the derivative nonlinear Schr\"odinger (DNLS) equation $iq_t(x,t)=-q_{xx}(x,t)+(r q^2)_x$ on the interval $[0,L]$. Assuming that the solution $q(x,t)$ exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\x$. This problem has explicit $(x,t)$ dependence, and it has jumps across $\{\x \in \C|\im{\x^4}=0 \}$. The relevant jump matrices are explicitly given in terms of the spectral functions $\{a(\x),b(\x)\},\{A(\x),B(\x)\}$, and $\{\ca({\x}),\cb(\x)\}$, which in turn are defined in terms of the initial data $q_0(x)=q(x,0)$, the boundary data $g_0(t)=q(0,t),g_1(t)=q_x(0,t)$, and another boundary values $f_0(t)=q(L,t),f_1(t)=q_x(L,t)$. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation., Comment: 30 pages. arXiv admin note: substantial text overlap with arXiv:0808.1534, arXiv:nlin/0412008

By introducing Lenard recursion equations, we derive a general coupled nonlinear Sch$\mathrm{\ddot{o}}$dinger (CNLS) hierarchy associated with well-known Manakov system and Sasa-Satsuma system. Based on the characteristic polynomial of Lax matrix for CNLS hierarchy, we obtain a third order algebraic curve $\mathcal{K}_{m-2}$ of arithmetic genus $m-2$, from which we establish the associated Baker-Ahhiezer functions, meromorphic function and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker-Ahhiezer functions, the meromorphic function, and in particular, of the algebro-geometric solutions for the entire CNLS hierarchy., Comment: We have found the results of this manuscript have been obtained by others

We provide a detailed treatment of Ruijsenaars-Toda (RT) hierarchy with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve hyperelliptic curve $\mathcal{K}_p$ associated with the Burchnall-Chaundy polynomial, Dubrovin-type equations for auxiliary divisors and associated trace formulas. With the help of a foundamental meromorphic function $\phi$, Baker-Akhiezer vector $\Psi$ on $\mathcal{K}_p$, the complex-valued algebro-geometric solutions of RT hierarchy are derived., Comment: 49 pages. arXiv admin note: substantial text overlap with arXiv:nlin/0702058, arXiv:nlin/0611055 by other authors

Though completely integrable Camassa-Holm (CH) equation and Degasperis-Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker-Akhiezer functions, meromorphic function and Dubrovin-type equations are established. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker-Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DP hierarchy., Comment: 65 pages. arXiv admin note: text overlap with arXiv:solv-int/9809004 by other authors

In this paper, based on the regular KdV system, we study negative order KdV (NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many conservation laws, and explicit multi-soliton and multi-kink wave solutions thorough bilinear B\"{a}cklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative order KdV hierarchy. The NKdV equations are not only gauge-equivalent to the Camassa-Holm equation through some hodograph transformations, but also closely related to the Ermakov-Pinney systems, and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The 1-kink wave solution is expressed in the form of $tanh$ while the 1-bell soliton is in the form of $sech$, and both forms are very standard. The collisions of 2-kink-wave and 2-bell-soliton solutions, are analyzed in details, and this singular interaction is a big difference from the regular KdV equation. Multi-dimensional binary Bell polynomials are employed to find bilinear formulation and B\"{a}cklund transformations, which produce $N$-soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasi-periodic wave solutions of the NKdV equations. Furthermore, the relations between quasi-periodic wave solutions and soliton solutions are clearly described. Finally, we show the quasi-periodic wave solution convergent to the soliton solution under some limit conditions., Comment: 61 pages, 4 figures

In this paper, we introduce a class of new generalized super Bell polynomials on a superspace, explore their properties, and show that they are a natural and effective tool to systematically investigate integrability of supersymmetric equations. The connections between the super Bell polynomials and super bilinear representation, bilinear B\"{a}cklund transformation, Lax pair and infinite conservation laws of supersymmetric equations are established. We take supersymmetric KdV equation and supersymmetric sine-Gordon equation to illustrate this procedure., Comment: 36 pages

Recently, Lembert, Gilson et al proposed a lucid and systematic approach to obtain bilinear B\"{a}cklund transformations and Lax pairs for constant-coefficient soliton equations based on the use of binary Bell polynomials. In this paper, we would like to further develop this method with new applications. We extend this method to systematically investigate complete integrability of nonisospectral and variable-coefficient equations. In addiction, a method is described for deriving infinite conservation laws of nonlinear evolution equations based on the use of binary Bell polynomials. All conserved density and flux are given by explicit recursion formulas. By taking variable-coefficient KdV and KP equations as illustrative examples, their bilinear formulism, bilinear B\"{a}cklund transformations, Lax pairs, Darboux covariant Lax pairs and conservation laws are obtained in a quick and natural manner. In conclusion, though the coefficient functions have influences on a variable-coefficient nonlinear equation, under certain constrains the equation turn out to be also completely integrable, which leads us to a canonical interpretation of their $N$-soliton solutions in theory., Comment: 39 pages