Integrable nonlinear evolution equations on the half-line.

*(English)*Zbl 1010.35089Summary: A rigorous methodology for the analysis of initial-boundary value problems on the half-line, \(0<x<\infty\), \(t>0\), is applied to the nonlinear Schrödinger (NLS), to the sine-Gordon (sG) in laboratory coordinates, and to the Korteweg-deVries (KdV) equation with dominant surface tension. Decaying initial conditions as well as a smooth subset of the boundary values \(\{\partial_x^l q(0,t) = g_l(t)\}^{n-1}_0\) are given, where \(n = 2\) for the NLS and the sG and \(n = 3\) for the KdV. For the NLS and the KdV equations, the initial condition \(q(x,0) = q_{0}(x)\) as well as one and two boundary conditions are given respectively; for the sG equation the initial conditions \(q(x,0) = q_{0}(x), q_{t}(x,0) = q_{1}(x)\), as well as one boundary condition are given. The construction of the solution \(q(x,t)\) of any of these problems involves two separate steps:

(a) Given decaying initial conditions define the spectral (scattering) functions \({a(k),b(k)}\). Associated with the smooth functions \(\{ g_l(t) \}^{n-1}_0\), define the spectral functions \({A(k),B(k)}\). Define the function \(q(x,t)\) in terms of the solution of a matrix Riemann-Hilbert problem formulated in the complex \(k\)-plane and uniquely defined in terms of the spectral functions \(\{a(k),b(k),A(k),B(k)\}\). Under the assumption that there exist functions \(\{g_l(t)\}^{n-1}_0\) such that the spectral functions satisfy a certain global algebraic relation, prove that the function \(q(x,t)\) is defined for all \(0<x<\infty\), \(t>0\), it satisfies the given nonlinear PDE, and furthermore that \(q(x,0) = q_0(x),\{\partial^l_xq(0,t) = g_l(t)\}^{n-1}_0\).

(b) Given a subset of the functions \(\{g_l(t)\}^{n-1}_0\) as boundary conditions, prove that the above algebraic relation characterizes the unknown part of this set. In general this involves the solution of a nonlinear Volterra integral equation which is shown to have a global solution. For a particular class of boundary conditions, called linearizable, this nonlinear equation can be bypassed and \(\{A(k),B(k)\}\) can be constructed using only the algebraic manipulation of the global relation. For the NLS, the sG, and the KdV, the following particular linearizable cases are solved: \(q_x(0,t)- \chi q(0,t) = 0\), \(q(0,t) = \chi\), \(\{ q(0,t) = \chi\), \(q_{xx}(0,t) = \chi+3\chi^2\}\), respectively, where \(\chi\) is a real constant.

(a) Given decaying initial conditions define the spectral (scattering) functions \({a(k),b(k)}\). Associated with the smooth functions \(\{ g_l(t) \}^{n-1}_0\), define the spectral functions \({A(k),B(k)}\). Define the function \(q(x,t)\) in terms of the solution of a matrix Riemann-Hilbert problem formulated in the complex \(k\)-plane and uniquely defined in terms of the spectral functions \(\{a(k),b(k),A(k),B(k)\}\). Under the assumption that there exist functions \(\{g_l(t)\}^{n-1}_0\) such that the spectral functions satisfy a certain global algebraic relation, prove that the function \(q(x,t)\) is defined for all \(0<x<\infty\), \(t>0\), it satisfies the given nonlinear PDE, and furthermore that \(q(x,0) = q_0(x),\{\partial^l_xq(0,t) = g_l(t)\}^{n-1}_0\).

(b) Given a subset of the functions \(\{g_l(t)\}^{n-1}_0\) as boundary conditions, prove that the above algebraic relation characterizes the unknown part of this set. In general this involves the solution of a nonlinear Volterra integral equation which is shown to have a global solution. For a particular class of boundary conditions, called linearizable, this nonlinear equation can be bypassed and \(\{A(k),B(k)\}\) can be constructed using only the algebraic manipulation of the global relation. For the NLS, the sG, and the KdV, the following particular linearizable cases are solved: \(q_x(0,t)- \chi q(0,t) = 0\), \(q(0,t) = \chi\), \(\{ q(0,t) = \chi\), \(q_{xx}(0,t) = \chi+3\chi^2\}\), respectively, where \(\chi\) is a real constant.

##### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q15 | Riemann-Hilbert problems in context of PDEs |

37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |