In this paper, we implement the Fokas method to study initial-boundary value problems of the mixed coupled nonlinear Schrödinger equation formulated on the half-line with Lax pairs involving 3×3 ...matrices. The solution can be written in terms of the solution to a 3×3 Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions s(k) and S(k), which are determined by the initial values and boundary values at x=0, respectively. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.
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→+∞ on the full-line.
We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve 3×3 matrices. Whereas initial value problems for integrable equations can be ...analyzed by means of the classical Inverse Scattering Transform (IST), the presence of a boundary presents new challenges. Over the last fifteen years, an extension of the IST formalism developed by Fokas and his collaborators has been successful in analyzing boundary value problems for several of the most important integrable equations with 2×2 Lax pairs, such as the Korteweg–de Vries, the nonlinear Schrödinger, and the sine-Gordon equations. In this paper, we extend these ideas to the case of equations with Lax pairs involving 3×3 matrices.
► We present a method for analyzing initial-boundary value problems for integrable equations. ► The equations have Lax pairs involving 3×3 matrices. ► The solution is given in terms of the solution of a Riemann–Hilbert problem. ► We show how to characterize the unknown boundary values.
The long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions at infinity is investigated by the nonlinear steepest descent method of Deift and Zhou. Three ...asymptotic sectors in space–time plane are found: the plane wave sector I, plane wave sector II and an intermediate sector with a modulated one-phase elliptic wave. The asymptotic solutions of the three sectors are proposed by successively deforming the corresponding Riemann–Hilbert problems to solvable model problems. Moreover, a time-dependent g-function mechanism is introduced to remove the exponential growths of the jump matrices in the modulated one-phase elliptic wave sector. Finally, the modulational instability is studied to reveal the criterion for the existence of modulated elliptic waves in the central region.
A class of Riemann–Hilbert problems on the real axis is formulated for solving the multicomponent AKNS integrable hierarchies associated with a kind of bock matrix spectral problems. Through special ...Riemann–Hilbert problems where a jump matrix is taken to be the identity matrix, soliton solutions to all integrable equations in each hierarchy are explicitly computed. A class of specific reductions of the presented integrable hierarchies is also made, together with its reduced Lax pairs and soliton solutions.
The inverse scattering transform is developed for a combined modified Korteweg–de Vrie equation through the technique of Riemann–Hilbert problems. From special Riemann–Hilbert problems with an ...identity jump matrix, soliton solutions are generated, which corresponds to the inverse scattering problems with reflectionless coefficients. A specific example of two-soliton solutions is explicitly presented, together with its 3d plots, contour plots and x-curve plots.
This paper presents a generalization of a recent iterative approach to solving a class of 2 × 2 matrix Wiener-Hopf equations involving exponential factors. We extend the method to square matrices of ...arbitrary dimension
, as arise in mixed boundary value problems with
junctions. To demonstrate the method, we consider the classical problem of scattering a plane wave by a set of collinear plates. The results are compared to other known methods. We describe an effective implementation using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.
Boundary value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order to study initial–boundary value problems ...of the general coupled nonlinear Schrödinger equation formulated on the finite interval with 3×3 Lax pairs. The solution can be written in terms of the solution of a 3×3 Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions s(k), S(k), and SL(k). The associated general Dirichlet to Neumann map is also analyzed via the global relation. It is interesting that the relevant formulas can be reduced to the analogous formulas derived for boundary value problems formulated on the half-line in the limit when the length of the interval tends to infinity. It is shown that the formulas characterizing the Dirichlet to Neumann map coincide with the analogous formulas obtained via a Gelfand–Levitan–Marchenko representation.
Recently, we have established the l2 bijectivity for the defocusing Ablowitz-Ladik system in the discrete weighted space l2,k with k∈N+ by the inverse spectral method. Based on these results, our ...study is to investigate the long-time asymptotics for the initial-value problem of the defocusing Ablowitz-Ladik system with initial potential in lower regularity without rapid decay. In this case, since the reflection coefficient is not smooth enough, it is impossible to directly apply the nonlinear steepest descent method. Our main idea is to transform the corresponding Riemann-Hilbert problem on the unit circle as the jump contour into a ∂¯-Riemann-Hilbert problem to be handled. As a result, we show that the solution admits the Zakharov-Manakov type formula in the region |n2t|≤V0<1, while the solution decays fast to zero in the region |n2t|≥V0>1.
The initial value problem for the Sasa–Satsuma equation is transformed to a 3×3 matrix Riemann–Hilbert problem with the help of the corresponding Lax pair. Two distinct factorizations of the jump ...matrix and a decomposition of the vector-valued function ρ(k) are given, from which the long-time asymptotics for the Sasa–Satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method.
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