We study the orbital stability and asymptotic stability problems for KdV solitons on the right half-line for nonhomogeneous boundary conditions in the energy space . This paper improves the results of Cavalcante and Mu & ntilde;oz [Revista Matem & aacute;tica Iberoamericana 35, no. 6 (2019); and SIAM Journal on Mathematical Analysis 55, no. 5 (2023): 4193-5992], which treat the homogeneous case. One of the key components of the stability argument in this work is the refinement of the estimate for the Lyapunov functional, given that the mass and energy, unlike in the context of homogeneous boundary conditions, do not exhibit a dissipative mechanism. As a consequence of this analysis, we obtain global control of the trace of the first and second derivatives of the solution to the model. This is, as far as we understand, the first orbital and asymptotic stability result for solitons posed on a half-line in the context of nonhomogeneous boundary conditions.

In this paper, we establish local well-posedness for the Cauchy problem associated with the Korteweg-de Vries (KdV) equation on a general metric star graph. The graph comprises m + k semi-infinite edges: k negative half-lines and m positive half-lines, all joined at a common vertex. The choice of boundary conditions is compatible with the conditions determined by the semigroup theory. The crucial point in this work is to obtain the integral formula using the forcing operator method and the Fourier restriction method of Bourgain. This work extends the results obtained by Cavalcante for the specific case of the Y junction to a more general class of star graphs.

We study special regularity properties of solutions to the initial-boundary value problem associated with the Korteweg-de Vries equations posed on the positive half-line. In particular, for initial data u(0 )is an element of H3+/4(R+) and boundary data f is an element of H3+/2(R+), where the restriction of u(0) to some subset of (b,infinity) has an extra regularity for any b > 0, we prove that the regularity of solutions u moves with infinite speed to its left as time evolves until a certain time T-& lowast;. The existence of a stopping time T-& lowast; appears because of the effect of the boundary function f. Also, as a consequence of our proof, we prove a gain in the regularity of the trace derivatives of the solutions for the Korteweg-de Vries on the half-line.

In this work, we establish local well-posedness for the Korteweg-de Vries model on a balanced star graph with a structure represented by semi-infinite edges, by considering a boundary condition of δ -type at the unique graph-vertex. Additionally, we extend the linear instability result in Angulo and Cavalcante (Nonlinearity 34:3373–3410, 2021) to one of nonlinear instability. For the proof of local well posedness theory, the principal new ingredient is the utilization of the special solutions by Faminskii in the context of half-lines. As far as we are aware, this approach is being used for the first time in the context of star graphs and can potentially be applied to other boundary classes. In the case of the nonlinear instability result, the principal ingredients are the linearized instability known result and the fact that data-to-solution map determined by the local theory is at least of class C^2 .

vanishing viscosity, networks. This work has received funding from the Alexander von Humboldt-Professorship program, the Transregio 154 Project "Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks" of the DFG, the grant PID2020-112617GB-C22 of MINECO (Spain), and the COST Action grant CA18232, "Mathematical models for interacting dynamics on networks" (MAT-DYN-NET). J. A. Barcena-Petisco is funded by the Grant PID2021-126813NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe", and by the grant IT1615-22 funded by the Basque Government. M. Cavalcante has been partially funded by CAPES-MATHAMSUD 88887.368708/2019. G. M. Coclite and N. De Nitti are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit & aacute; e le loro Applicazioni (GNAMPA) of the 2, Investment 1.4 (Call for tender No. 3138 of 16/12/2021), of Italian Ministry of University and

In this work we are concerned with solutions to the linear Schrödinger type system with mixed dispersion, the so-called biharmonic Schrödinger equation. Precisely, we are able to prove an exact control property for these solutions with the control in the energy space posed on an oriented star graph structure \begin{document}$ \mathcal{G} $\end{document} for \begin{document}$ T>T_{min} $\end{document}, with \begin{document}$ T_{min} = \sqrt{ \frac{ \overline{L} (L^2+\pi^2)}{\pi^2\varepsilon(1- \overline{L} \varepsilon)}}, $\end{document} when the couplings and the controls appear only on the Neumann boundary conditions.

In this paper we study the asymptotic stability problem for KdV solitons on the half-line, with zero boundary condition and absence of the drift term, represented as u(x). Unlike standard KdV, these are not exact solutions to the equation. In a previous result, we showed that these solitons are orbitally stable, provided they are placed sufficiently far from the origin. In this paper, we prove their asymptotic stability in the energy space, and provide decay properties for all remaining regions, except the "small soliton region". For the proof we follow the ideas by Martel and Merle for the big soliton part, and for the linearly dominated region we follow recent results on generalized KdV decay.

In this paper we study the stability problem for mKdV breathers on the left half-line. We are able to show that leftwards moving breathers, initially located far away from the origin, are strongly stable for the problem posed on the left half-line, when assuming homogeneous boundary conditions. The proof involves a Lyapunov functional which is almost conserved by the mKdV flow once we control some boundary terms which naturally arise.

In a recent article [16], the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [16] concerning the study of the cubic fourth order Schrodinger equation in a star graph structure G. Precisely, consider G composed by N edges parameterized by half-lines (0, + infinity) attached with a common vertex nu. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the boundary forcing operator approach. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.

The aim of this work is to establish a novel linear instability criterion for the Korteweg-de Vries (KdV) model on metric graphs. In the case of balanced graphs with a structure represented by a finite collection of semi-infinite edges and with boundary condition of delta-type interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable. In this case, the use of the analytic perturbation theory of operators as well as the extension theory of symmetric operators is fundamental in our stability analysis. The arguments showed in this investigation have prospects in the study of the instability of stationary waves solutions for nonlinear evolution equations on metric graph.

In this work we study the initial boundary value problem associated with the coupled Schrödinger equations with quadratic nonlinearities, that appears in nonlinear optics, on the half-line. We obtain local well-posedness for data in Sobolev spaces with low regularity, by using a forcing problem on the full line with a presence of a forcing term in order to apply the Fourier restriction method of Bourgain. The crucial point in this work is the new bilinear estimates on the classical Bourgain spaces X with b < 1 2 , jointly with bilinear estimates in adapted Bourgain spaces that will used to treat the traces of nonlinear part of the solution. Here the understanding of the dispersion relation is the key point in these estimates, where the set of regularity depends strongly of the constant a measures the scaling-diffraction magnitude indices.

This paper concerns the initial-boundary value problem of the Kawahara equation posed on the right and left half-lines. We prove the local well-posedness in the low regularity Sobolev space. We introduce the Duhamel boundary forcing operator, which is introduced by Colliander and Kenig (Commun Partial Differ Equ 27:2187–2266, 2002) in the context of Airy group operators, to construct solutions on the whole line. We also give the bilinear estimate in $$X^{s,b}$$ space for $$b < \frac{1}{2}$$, which is almost sharp compared to IVP of Kawahara equation (Chen et al. in J Anal Math 107:221–238, 2009; Jia and Huo in J Differ Equ 246:2448–2467, 2009).

The initial–boundary value problem for the Schrödinger–Korteweg–de Vries system is considered on the left and right half-lines for a wide class of initial–boundary data, including the energy regularity $$H^1({\mathbb {R}}^{\pm })\times H^1({\mathbb {R}}^{\pm })$$ for initial data. Assuming homogeneous boundary conditions, for the problem on the positive half-line, it is shown for positive coupling interactions that local solutions can be extended globally in time for initial data in the energy space. Furthermore, for negative coupling interactions, for a certain class of regular initial data, the following result was proved: if the respective solution does not exhibit finite-time blow-up in $$H^1({\mathbb {R}}^-)\times H^1({\mathbb {R}}^-)$$, then the norm of the weighted space $$L^2\big ({\mathbb {R}}^-,\, |x|\mathrm{d}x\big )\times L^2\big ({\mathbb {R}}^-,\, |x|\mathrm{d}x\big )$$ blows up at infinity time with super-linear rate; this is obtained by using a satisfactory algebraic manipulation of a new global virial-type identity associated with the system, which does not work in the context of whole real line.

This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schr\"odinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces introducing Duhamel boundary forcing operator associated to the linear equation to construct solutions on the whole line. With this in hands, the energy and nonlinear estimates allow us to apply Fourier restriction method, introduced by J. Bourgain, to get the main result of the article. Additionally, adaptations of this approach for the biharmonic cubic nonlinear Schr\"odinger equation on star graphs are also discussed.

In the last years the study of initial boundary value problems for nonlinear dispersive equations on the half-lines has given attention of many researchers. This turns out to be a rather challenging problem, mainly when studied in low Sobolev regularity. In this note we present a review of the main results about this topic and also introduce interesting open problems which still requires attention from the mathematical point of view.

In this paper we study the stability problem for KdV solitons on the left and right half-line. Unlike standard KdV, these are not exact solutions to the equations posed in the half-line. However, we are able to show that solitons placed initially far away from the origin are strongly stable for the problem posed on the right half-line, assuming homogeneous boundary conditions. For the problem posed on the left half-line, the positive infinitetime stability problem makes no sense for the case of KdV solitons, but in this setting we prove a result of stability for all negative times. The proof involves the use of almost conserved quantities adapted to the evolution of the KdV soliton on the particular case of the half-line. Adaptations to other boundary conditions or star graphs are also discussed.

We prove local well-posedness for the initial-boundary value problem (IBVP) associated to the Schrödinger–Korteweg–de Vries system on right and left half-lines. The results are obtained in the low regularity setting by using two analytic families of boundary forcing operators, one of these families being developed by Holmer to study the IBVP associated to the Korteweg–de Vries equation [The initial-boundary value problem for the Korteweg–de Vries equation, Comm. Partial Differential Equations 31 (2006) 1151–1190] and the other one was recently introduced by Cavalcante [The initial-boundary value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential Integral Equations 30(7–8) (2017) 521–554] in the context of nonlinear Schrödinger with quadratic nonlinearities.

This paper is a continuation of authors' previous work [6]. We extend the argument [6] to fifth-order KdV-type equations with different non-linearities, in specific, where the scaling argument does not hold. We establish the X-s,X-b nonlinear estimates for b < 1/2, which is almost optimal compared to the standard X-s,X-b nonlinear estimates for b > 1/2 [8, 17]. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.

The main purpose of this paper is to show the global stabilization and exact controllability properties of a fourth order nonlinear Schrödinger system on a periodic domain \(\mathbb {T}\) with internal control supported on an arbitrary sub-domain of \(\mathbb {T}\). More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exact controllability shows that fourth order nonlinear Schrödinger is globally exactly controllable.

This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schrödinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces introducing Duhamel boundary forcing operator associated to the linear equation to construct solutions on the whole line. With this in hands, the energy and nonlinear estimates allow us to apply Fourier restriction method, introduced by J. Bourgain, to get the main result of the article. Additionally, adaptations of this approach for the biharmonic cubic nonlinear Schrödinger equation on star graphs are also discussed.