Finite difference scheme for a high order nonlinear Schrodinger equation with localized damping

Cavalcanti, Marcelo M.; Corrêa, Wellington José; Sepúlveda, Carolina; Asem, Rodrigo Véjar

doi:10.24193/subbmath.2019.2.03

Cited by 4 publications

(9 citation statements)

References 16 publications

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“…and S(t)w 0 denotes the solution of the corresponding linear equation (8). By using the linear homogeneous and nonhomogeneous estimates, we obtain that for any z ∈ Y T , one has…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…A graph and a contour plot of the kernel are given below in Figure 2 for the particular values of parameters given by L = π, β = 1, α = 2, δ = 8, and r = 1. . = e rt w(x, t), where r > 0 and w is the sought-after solution of the linearized target system (8). Thenw satisfies the following pde model…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…One can see this by multiplying (8) withw, integrating over (0, L) and taking the imaginary parts. It is clear from (7) and the boundary conditions w(0, t) = 0, u(0, t) = g 0 (t) that the backstepping feedback must have the form g 0 (t) .…”

mentioning

confidence: 99%

“…The difficulty is generally associated with finding a suitable kernel k so that one can reach at the target system (8) starting from the original plant (6). Once such kernel is found and one proves that the backstepping transformation is bounded invertible on a suitable space, then one can conclude that the same decay rate property also holds for the solution of (6).…”

mentioning

confidence: 99%

“…Regarding the controllability aspect, the internal stabilization of the HNLS with constant coefficients was studied by [11] and [4]. A numerical treatment of this problem was given in [8]. The exact boundary controllability for the higher order nonlinear Schrödinger equation with constant coefficients was studied in [9].…”

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confidence: 99%

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Stabilization of higher order Schrödinger equations on a finite interval: Part I

Batal¹,

Özsarı²,

Yılmaz³

2021

EECT

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We study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation.

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“…and S(t)w 0 denotes the solution of the corresponding linear equation (8). By using the linear homogeneous and nonhomogeneous estimates, we obtain that for any z ∈ Y T , one has…”

Section: Preliminaries and Notationmentioning

confidence: 99%

Section: Preliminaries and Notationmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Stabilization of higher order Schrödinger equations on a finite interval: Part I

Batal¹,

Özsarı²,

Yılmaz³

2021

EECT

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Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

Alkın,

Mantzavinos,

Özsarı

2023

Stud Appl Math

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We establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator.

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Stabilization of higher order Schrödinger equations on a finite interval: Part II

Özsarı

Yılmaz

2022

EECT

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<p style='text-indent:20px;'>Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [<xref ref-type="bibr" rid="b3">3</xref>] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.</p>

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Finite difference scheme for a high order nonlinear Schrodinger equation with localized damping

Cited by 4 publications

References 16 publications

Stabilization of higher order Schrödinger equations on a finite interval: Part I

Stabilization of higher order Schrödinger equations on a finite interval: Part I

Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

Stabilization of higher order Schrödinger equations on a finite interval: Part II

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