The Unified Method: III Non-Linearizable Problems on the Interval

Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving six scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other four depend on all boundary values. The most difficult step of the new method is the characterization of the latter four spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data. We present two different characterizations of this problem. One is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions. The other is based on the analysis of the global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the relevant spectral functions. We also show that these two different characterizations are equivalent and that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained recently for the case of boundary value problems formulated on the half-line.Comment: 22 page

On a novel integrable generalization of the nonlinear Schr\"odinger equation

We consider an integrable generalization of the nonlinear Schr\"odinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.Comment: 20 pages, 1 figur

Long-time asymptotics for the Degasperis-Procesi equation on the half-line

We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula for the leading order asymptotics of the solution in the similarity region in terms of the initial and boundary values.Comment: 61 pages, 11 figure

On Boussinesq's equation for water waves

A century and a half ago, J. Boussinesq derived an equation for the propagation of shallow water waves in a channel. Despite the fundamental importance of this equation for a number of physical phenomena, mathematical results on it remain scarce. One reason for this is that the equation is ill-posed. In this paper, we establish several results on the Boussinesq equation. First, by solving the direct and inverse problems for an associated third-order spectral problem, we develop an Inverse Scattering Transform (IST) approach to the initial value problem. Using this approach, we establish a number of existence, uniqueness, and blow-up results. For example, the IST approach allows us to identify physically meaningful global solutions and to construct, for each $T > 0$, solutions that blow up exactly at time $T$. Our approach also yields an expression for the solution of the initial value problem for the Boussinesq equation in terms of the solution of a Riemann--Hilbert problem. By analyzing this Riemann--Hilbert problem, we arrive at asymptotic formulas for the solution. We identify ten main asymptotic sectors in the $(x,t)$-plane; in each of these sectors, we compute an exact expression for the leading asymptotic term together with a precise error estimate. The asymptotic picture that emerges consists, roughly speaking, of two nonlinearly coupled copies of the corresponding picture for the (unidirectional) KdV equation, one copy for right-moving and one for left-moving waves. Of particular interest are the sectors which describe the interaction of right and left moving waves, which present qualitatively new phenomena.Comment: 111 pages, 23 figure

The Unified Method: I Non-Linearizable Problems on the Half-Line

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions.Comment: 39 page