Integrable Nonlinear Evolution Equations on a Finite Interval

Abstract

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.

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

“…where W j is given by (12) with μ replaced with μ j , and the contours {γ j } 4 1 are shown in Figure 1. The first, second, and third column of the matrix equation (13) involves the exponentials And we have the following inequalities on the contours:…”

“…and { l1 (t, k)} 3 l=1 , { l2 (t, k)} 3 l=1 satisfy the following system of integral equations: 12 (t, k) = t 0 e −4ik 2 (t−t ) − iσ (|g 01 | 2 + |g 02 | 2 ) 12 + (2kg 01 + ig 11 ) 22…”

“…The Fokas method provides a generalization of the IST formalism from initial value to IBV problems, and over the last 18 years, this method has been used to analyze boundary value problems for several of the most important integrable equations with 2 × 2 Lax pairs, such as the Kortewegde Vries [7], the nonlinear Schrödinger [8], the sine-Gordon equations [9], see [10,11,12,13], or see a series of papers [14,15,16]. Just like the IST on the line, the unified method yields an expression for the solution of an IBV problem in terms of the solution of a Riemann-Hilbert problem.…”

The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method

“…[12][13][14][15][16][17][18][19][20][21][22][23][24]. The Fokas method provides an expression for the solution of an initial-boundary value (IBV) problem in terms of the solution of an RHP.…”

Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane

“…Following the many successes of the inverse scattering approach, one of the main open problems in the area of integrable systems in the late twentieth century was the extension of the IST formalism to initial-boundary value (IBV) problems, see [1]. Such an extension was introduced by Fokas in [3] (see also [4,5]) and has subsequently been developed and applied by several authors [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In analogy with the IST on the line, the unified transform of [3] relies for the analysis of an IBV problem on the definition of several spectral functions via nonlinear Fourier transforms and on the formulation of a RH problem.…”