Abstract:
The unified transform method introduced by Fokas can be used to analyze initial-boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann-Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of $L^2$-RH problems, we consider the construction of quarter plane solutions which are $C^1$ in time and $C^3$ in space.

Abstract:
This article concerns an initial-boundary value problem in a quarter-plane for the Korteweg-de Vries (KdV) equation. For general nonlinear boundary conditions we prove the existence and uniqueness of a global regular solution.

Abstract:
A global bifurcation theorem for the following nonlinear Sturm–Liouville problem is given Moreover we give various versions of existence theorems for boundary value problems The main idea of these proofs is studying properties of an unbounded connected subset of the set of all nontrivial solutions of the nonlinear spectral problem , associated with the boundary value problem , in such a way that .

Abstract:
By using the M${\rm \ddot{o}}$nch fixed point theorem,the existence of nonlinear Sturm-Liouville problems in Banach spaceis investigated under more general conditions than those used in literatures.

Abstract:
This paper concerns the nonlinear Sturm-Liouville problem $$ -u''(t) + f(u(t)) = lambda u(t), quad u(t) > 0, quad t in I := (0, 1), quad u(0) = u(1) = 0, $$ where $lambda $ is a positive parameter. We try to determine the nonlinear term $f(u)$ by means of the global behavior of the bifurcation branch of the positive solutions in $mathbb{R}_+ imes L^2(I)$.

Abstract:
In this paper the Adomian decomposition method is applied to the nonlinear Sturm-Liouville problem -y" + y(t)p=λy(t), y(t) > 0, t ∈ I = (0, 1), y(0) = y(1) = 0, where p > 1 is a constant and λ > 0 is an eigenvalue parameter. Also, the eigenvalues and the behavior of eigenfuctions of the problem are demonstrated.

Abstract:
We consider the rigorous derivation of asymptotic formulas for initial-boundary value problems using the nonlinear steepest descent method. We give detailed derivations of the asymptotics in the similarity and self-similar sectors for the mKdV equation in the quarter plane. Precise and uniform error estimates are presented in detail.

By mixed monotone method, we establish the existence and uniqueness of
positive solutions for fourth-order nonlinear singular Sturm-Liouville
problems. The theorems obtained are very general and complement previously
known results.

Abstract:
Dromions are exponentially localised coherent structures supported by nonlinear integrable evolution equations in two spatial dimensions.In the study of initial-value problems on the plane, such solutions occur only if one imposes nontrivial boundary conditions at infinity, a situation of dubious physical significance. However it is established here that dromions appear naturally in the study of boundary-value problems. In particular, it is shown that the long time asymptotics of the solution of the Davey-Stewartson I equation in the quarter plane with arbitrary initial conditions and with zero Dirichlet boundary conditions is dominated by dromions. The case of non-zero Dirichlet boundary conditions is also discussed.

Abstract:
We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $\omega$-parametric analytic family better than order of $(\omega\ln\omega)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $\omega^{-m}$.