Acta mathematica scientia, Series B >
A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION
Received date: 2013-03-21
Revised date: 2014-01-04
Online published: 2014-07-20
Supported by
The work described in this paper was supported by grants from the National Science Foundation of China (10971031; 11271079; 11075055), Doctoral Programs Foundation of the Ministry of Education of China, and the Shanghai Shuguang Tracking Project (08GG01).
We use the Fokas method to analyze the derivative nonlinear Schr¨odinger (DNLS) equation iqt(x, t) = −qxx(x, t)+(rq2)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 ξ. This problem has explicit (x, t) dependence, and it has jumps across {ξ ∈ C|Imξ4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a(ξ), b(ξ)}, {A(ξ), B(ξ)}, and {A(ξ), B(ξ)}, which in turn are defined in terms of the initial data q0(x) = q(x, 0), the boundary data g0(t) = q(0, t), g1(t) = qx(0, t), and another boundary values f0(t) = q(L, t), f1(t) =qx(L, t). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.
XU Jian , FAN En-Gui . A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION[J]. Acta mathematica scientia, Series B, 2014 , 34(4) : 973 -994 . DOI: 10.1016/S0252-9602(14)60063-1
[1] Fokas A S. A unified approach to boundary value problems//CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 2008
[2] Fokas A S. A unified transform method for solving linear and certain nonlinear PDEs. Proc R Soc Lond A, 1997, 453: 1411–1443
[3] Fokas A S. Integrable nonlinear evolution equations on the half-line. Commun Math Phys, 2002, 230: 1–39
[4] Fokas A S, Its A R, Sung L Y. The nonlinear Schr¨odinger equation on the half-line. Nonlinearity, 2005, 18: 1771–1822
[5] Fokas A S, Its A R. The linearization of the initial-boundary value problem of the nonlinear Schr¨odinger equation. SIAM J Math Anal, 1996, 27: 738–764
[6] Boutet de Monvel A, Fokas A S, Shepelsky D. The mKDV equation on the half-line. J Inst Math Jussieu, 2004, 3: 139–164
[7] Boutet de Monvel A, Shepelsky D. Initial boundary value problem for the MKdV equation on a finite interval. Ann Inst Fourier (Grenoble), 2004, 54: 1477–1495
[8] Fokas A S, Its A R. The Nonlinear Schr¨odinger equation on the interval. J Phys A, 2004, 37: 6091–6114
[9] Lenells J. The derivative nonlinear Schr¨odinger equation on the half-line. Physica D 2008, 237: 3008–3019
[10] Mjolhus E. On the modulational instability of hydromagnetic waves parallel to the magnetic field. J Plasma Phys, 1976, 16: 321–334
[11] Kodama Y. Optical solitons in a monomode fiber. J Stat Phys, 1985, 35: 597–614
[12] Kaup D J, Newell A C. An exact solution for a derivative nonlinear Schr¨odinger equation. J Math Phys, 1978, 19: 789–801
[13] Kawata T, Inoue H. Exact solutions of the derivative nonlinear Schr¨odinger equation under the nonvanishing
conditions. J Phys Soc Japan, 1978, 44: 1968–1976
[14] Ma W X, Zhou R G. On inverse recursion operator and tri-Hamiltonian formulation for a Kaup-Newell system of DNLS equations. J Phys A, 1999, 32: L239–L242
[15] Ichikawa Y H, Konno K, Wadati M, et al. Spiky soliton in circular polarized Alfve wave. J Phys Soc Japan, 1980, 48: 279–286
[16] Lashkin V M. N-soliton solutions and perturbation theory for the derivative nonlinear Schr¨odinger equation with nonvanishing boundary conditions. J Phys A, 2007, 40: 6119–6132
[17] Xu S W, He J S, Wang L H. The Darboux transformation of the derivative nonlinear Schr¨odinger equation. J Phys A, 2011, 44: 305203-305225
[18] Fan E G. Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation. J Phys A, 2000, 33: 6925–6933
[19] Fokas A S. Two dimensional linear PDEs in a convex ploygon. Proc R Soc Lond A, 2001, 457: 371–393
[20] Adler V E, G¨urel B, G¨urses M, et al. Boundary conditions for integrable equations. J Phys A, 1997, 30: 3505–3513
[21] Zakharov V E, Shabat A. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, I and II. Funct Anal Appl, 1974, 8: 226–235; 1979, 13: 166–174
[22] Chen H H, Lee Y S, Liu C S. Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys Scr, 1979, 20: 490–492
[23] Kundu A, Strampp W, Oevel W. Gauge transformations of constrained KP flows: new integrable hierarchies. J Math Phys, 1995, 36: 2972–2984
[24] Fan E G. A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations. J Math Phys, 2001, 42: 4327–4344
[25] Beals R, Coifman R. Scattering and inverse scattering for first order systems. Comm in Pure and Applied Math, 1984, 37: 39–90
[26] Deift D, Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems. Ann Math, 1993, 137(2): 295–368
[27] Chen X-J, Lam W K. Inverse scattering transform for the derivative nonlinear Schrodinger equation with nonvanishing boundary conditions. Phys Rev E, 2004, 69: 066604
[28] Cai H, Huang N-N. The Hamiltonian formalism of the DNLS equation with a nonvanished boundary value. J Phys A: Math Gen, 2006, 39: 5007–14
[29] Steudel H. The hierarchy of multi-soliton solutions of the derivative nonlinear Schr¨odinger equation. J Phys A: Math Gen, 2003, 36: 1931–46
[30] Xu Shuwei, He Jingsong, Wang Lihong. Two kinds of rogue waves of the general nonlinear Schrodinger equation with derivative. Europhys Lett, 2012, 97: 30007, 6pp
[31] Hitzazis I, Tsoubelis D. The Korteweg-de Vries equation on the interval. J Math Phys, 2010, 51: 083520. 1–32
[32] Agrawal G P. Nonlinear Fiber Optics. Academic Press, 2007
/
| 〈 |
|
〉 |