Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

Monvel, Anne Boutet de; Fokas, A. S.; Shepelsky, Dmitry

doi:10.1023/b:math.0000010711.66380.77

Cited by 80 publications

(130 citation statements)

References 4 publications

Supporting

Mentioning

126

Contrasting

Unclassified

Order By: Relevance

“…For example, for the Dirichlet problem for the NLS equation on the half-line, the method of [3] yields q x (0, t) in terms of q(x, 0) and q(0, t). Actually, it is shown in [8] and [9] that q x (0, t) can be expressed explicitly through the solution of a system of nonlinear ODEs uniquely defined in terms of q(x, 0) and q(0, t). This is the first time in the literature that such an explicit result is obtained for a nonlinear evolution PDE.…”

Section: Discussionmentioning

confidence: 99%

“…The functions L j and M j are simply expressed in terms of the matrix entries of L and M . For the NLS equation, the corresponding representation is presented in details in [8]. For the sG equation, the details are given in [9].…”

Section: Analysis Of the Global Relationmentioning

confidence: 99%

“…Although rigorous results in this direction were obtained in [5], the relevant formalism is quite complicated. A dramatic simplification was announced in [8] where it was shown that the global relation can be effectively analyzed if one introduces a Gelfand-LevitanMarchenko representation for the eigenfunction of the t-part of the Lax pair evaluated at x = 0. If this eigenfunction is denoted by (Φ 1 (t, k), Φ 2 (t, k))…”

Section: Introductionmentioning

confidence: 99%

“…T , and if the functions involved in its Gelfand-LevitanMarchenko representation are denoted by {L j ,M j } 2 1 , then it is shown in [8] that, in the case of the Dirichlet problem for the NLS equation, q x (0, t) can be explicitly expressed in terms of {L j ,M j } 2 1 and of the initial and boundary conditions (q(x, 0) and q(0, t)). This yields q x (0, t) in terms of a system of four nonlinear ODEs satisfied by the functions {L j ,M j } 2 1 .…”

Section: Introductionmentioning

confidence: 99%

“…However, the relevant global relation was not analyzed in this paper. The analogous RH problem for the NLS equation together with the analysis of the global relation is presented in [11]; the latter analysis is based on the results of [8].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Integrable Nonlinear Evolution Equations on a Finite Interval

Monvel

Fokas

Shepelsky

2006

Commun. Math. Phys.

View full text Add to dashboard Cite

Let q(x, t) satisfy an integrable nonlinear evolution PDE on the interval 0 < x < L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x = 0 and n − N boundary conditions at x = L, where if n is even, N = n/2, and if n is odd, N is either (n − 1)/2 or (n + 1)/2, depending on the sign of ∂ n x q. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving qt + qxxx and qt − qxxx one must prescribe one and two boundary conditions, respectively, at x = 0. We will refer to these two mKdV equations as mKdV I and mKdV II, respectively.Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV I and mKdV II equations. We first show that the unknown boundary values at each end (for example, qx(0, t) and qx(L, t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. For the sG and the focusing versions of mKdV I and mKdV II equations, this system has a global solution, while for the defocusing versions of mKdV I and mKdV II equations, the global existence remains open. We then show that q(x, t) can be expressed in terms of the solution of a 2 × 2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x, t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp{i(k − 1/k)x + i(k + 1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k), b(k)}, {A(k), B(k)}, and {A(k), B(k)}, which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x = 0, and of the boundary values of q and of its x-derivatives at x = L, respectively. This Riemann-Hilbert problem has a global solution.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Analysis Of the Global Relationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Integrable Nonlinear Evolution Equations on a Finite Interval

Monvel

Fokas

Shepelsky

2006

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs

Fokas

2005

Comm. Pure Appl. Math.

Self Cite

103

127

View full text Add to dashboard Cite

Let q(x, t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half-line for such a PDE is at least linearly well-posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n 2 and for n odd, depending on the sign of the highest derivative, N equals either n−1 2 or n+1 2 . For example, for the nonlinear Schrödinger (NLS) and the sine-Gordon (sG), N = 1, while for the modified Korteweg-deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet-to-Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized.

show abstract