arXiv subjects

Dionyssios Mantzavinos

Publications and source records attributed to Dionyssios Mantzavinos.

25 records · Page 2Linked to original sources

arXiv2020

The Linearized Classical Boussinesq System on the Half-Line

The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.

math.AP↗

arXiv2020

Polynomial Bound and Nonlinear Smoothing for the Benjamin-Ono Equation on the Circle

For initial data in Sobolev spaces $H^s(\mathbb T)$, $\frac 12 < s \leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-\frac 12) + ε}$, $0<ε\ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.

math.AP↗

arXiv2018

Well-Posedness of Initial-Boundary Value Problems for a Reaction-Diffusion Equation

A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The result is established via a contraction mapping argument, taking advantage of a novel approach that utilizes the formula produced by the unified transform method of Fokas for the forced linear heat equation to obtain linear estimates analogous to those previously derived for the nonlinear Schrödinger, Korteweg-de Vries and "good" Boussinesq equations. Thus, the present work extends the recently introduced "unified transform method approach to well-posedness" from dispersive equations to diffusive ones.

math.AP↗

arXiv2018

Well-Posedness of the Nonlinear Schrödinger Equation on the Half-Plane

The initial-boundary value problem (IBVP) for the nonlinear Schrödinger (NLS) equation on the half-plane with nonzero boundary data is studied by advancing a novel approach recently developed for the well-posedness of the cubic NLS on the half-line, which takes advantage of the solution formula produced by the unified transform of Fokas for the associated linear IBVP. For initial data in Sobolev spaces on the half-plane and boundary data in Bourgain spaces arising naturally when the linear IBVP is solved with zero initial data, the present work provides a local well-posedness result for NLS initial-boundary value problems in higher dimensions.

math.AP↗

arXiv2018

Soliton trapping, transmission and wake in modulationally unstable media

Interactions between solitons and the coherent oscillation structures generated by localized disturbances via modulational instability are studied within the framework of the focusing nonlinear Schrodinger equation. Two main interaction regimes are identified based on the relative value of the velocity of the incident soliton compared to the amplitude of the background: soliton transmission and soliton trapping. Specifically, when the incident soliton velocity exceeds a certain threshold, the soliton passes through the coherent structure and emerges on the other side with its velocity unchanged. Conversely, when the incident soliton velocity is below the threshold, once the soliton enters the coherent structure, it remains confined there forever. It is demonstrated that the soliton is not destroyed, but its velocity inside the coherent structure is different from its initial one. Moreover, it is also shown that, depending on the location of the discrete eigenvalue associated to the soliton, these phenomena can also be accompanied by the generation of additional, localized propagating waves in the coherent structure, akin to a soliton-generated wake.

nlin.PS↗

arXiv2015

Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability

The long-time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. First, the IST is formulated over a single sheet of the complex plane without introducing a uniformization variable. The solution of the focusing NLS equation with nonzero boundary conditions is thus associated with a suitable matrix Riemann-Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well-known modulational instability within the context of the IST. This growth is then removed by suitable deformations of the Riemann-Hilbert problem in the complex spectral plane. Asymptotically in time, the $xt$-plane is found to decompose into two types of regions: a left far-field region and a right far-field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. In the latter region, it is also shown that the modulus of the leading order solution, which is initially obtained in the form of a ratio of Jacobi theta functions, eventually reduces to the well-known elliptic solution of the focusing NLS equation. These results provide the first characterization of the long-time behavior of generic perturbations of a constant background in a modulationally unstable medium.

math.AP↗

arXiv2011

The Kadomtsev-Petviashvili I Equation on the Half-Plane

A new method for the solution of initial-boundary value problems for \textit{linear} and \textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 \cite{F1997}. This approach was subsequently extended to initial-boundary value problems for evolution PDEs in two spatial dimensions, first in the case of linear PDEs \cite{F2002b} and, more recently, in the case of integrable nonlinear PDEs, for the Davey-Stewartson and the Kadomtsev-Petviashvili II equations on the half-plane (see \cite{FDS2009} and \cite{MF2011} respectively). In this work, we study the analogous problem for the Kadomtsev-Petviashvili I equation; in particular, through the simultaneous spectral analysis of the associated Lax pair via a d-bar formalism, we are able to obtain an integral representation for the solution, which involves certain transforms of all the initial and the boundary values, as well as an identity, the so-called global relation, which relates these transforms in appropriate regions of the complex spectral plane.

math.AP↗