arXiv subjects

Engui Fan

Publications and source records attributed to Engui Fan.

At least 73 records · Page 4Linked to original sources

arXiv2019

Riemann-Hilbert approach to the modified nonlinear Schröinger equation with non-vanishing asymptotic boundary conditions

The modified nonlinear Schrödinger (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.

nlin.SI↗

arXiv2019

The Sasa-Satsuma equation with non-vanishing boundary conditions

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.

nlin.SI↗

arXiv2019

Inverse scattering transformation for the Fokas-Lenells equation with nonzero boundary conditions

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.

nlin.SI↗

arXiv2019

Long-time asymptotic behavior of the modified Schrödinger equation via Dbar-steepest descent method

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}).

nlin.SI↗

arXiv2019

Long time asymptotics behavior of the focusing nonlinear Kundu-Eckhaus equation

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 $\mathcal{O}(t^{-3/4})$.

nlin.SI↗

arXiv2019

Long-time asyptotics behavior for the integrable modified Camassa-Holm equation with cubic nonlinearity

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.

nlin.SI↗

arXiv2019

The Riemann-Hilbert approach to focusing Kundu-Eckhaus equation with nonzero boundary conditions

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.

nlin.SI↗

arXiv2018

Long-time asymptotics for the Nonlocal mKdV equation

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.

nlin.SI↗

arXiv2018

Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $α=β$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $α=-1/2,$ and $α=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^β$ and $x^{-1/2}(1-x)^β$ The $σ$ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto $σ$ form of $P_{V}$ and $P_{VI}$, although in the shift Jacobi case, with the weight $x^α(1-x)^β,$ the $β$ parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.

math-ph↗

arXiv2017

Hankel determinants for a perturbed Laguerre weight and Painleve V equation

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.

nlin.SI↗

arXiv2017

Long-time asymptotic behavior for the complex short pulse equation

In this paper, we consider the initial value problem for the complex short pulse equation with a Wadati-Konno-Ichikawa type Lax pair. We show that the solution to the initial value problem has a parametric expression in terms of the solution of $2\times 2$-matrix Riemann-Hilbert problem, from which an implicit one-soliton solution is obtained on the discrete spectrum. While on the continuous spectrum we further establish the explicit long-time asymptotic behavior of the non-soliton solution by using Deift-Zhou nonlinear steepest descent method.

math-ph↗

arXiv2017

Critical edge behavior in the perturbed Laguerre ensemble and the Painleve V transcendent

In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of $$w(x,t)=(x+t)^λx^αe^{-x}$$ with $ x\geq 0,\ t>0,\ α>0,\ α+λ+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 $φ$-function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlevé V (shorted as P$_{\rm V}$) transcendent via a simple transformation. Moreover, this P$_{\rm V}$ transcendent is equivalent to a general Painlevé P$_{\rm III}$ transcendent. For large $s,$ the P$_{\rm V}$ kernel reduces to the Bessel kernel $\mathbf{J}_{α+λ}.$ For small $s,$ the P$_{\rm V}$ kernel reduces to another Bessel kernel $\mathbf{J}_α.$ At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.

math-ph↗

arXiv2017

Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

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 $s(λ)$, $S(λ)$ and $S_L(λ)$, 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.

nlin.SI↗

arXiv2015

The unified transform method for the Sasa-Satsuma equation on the interval

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.

nlin.SI↗

arXiv2015

The GLM representation of the global relation for the two-component nonlinear Schrödinger equation on the interval

In a previous work, we show that the solution of the initial-boundary value problem for the two-component nonlinear Schrödinger 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.

nlin.SI↗

arXiv2015

Initial-boundary value problem for integrable nonlinear evolution equations with $3\times 3$ Lax pairs on the interval

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.

nlin.SI↗

arXiv2015

The Ostrovsky-Vakhnenko equation on the half-line: a Riemann-Hilbert approach

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.

nlin.SI↗

arXiv2015

Large n-limit for Random matrices with External Source with 3 eigenvalues

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.

math-ph↗