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1. Integrable nonlocal derivative nonlinear Schrödinger equations

   https://doi.org/10.1088/1361-6420/ac5f75  

   Ablowitz, Mark J; Luo, Xu-Dan; Musslimani, Ziad H; Zhu, Yi (April 2022, Inverse Problems)

   Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation. 

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2. On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

   https://doi.org/10.1090/cams/32  

   Lührmann, Jonas; Schlag, Wilhelm (February 2024, Communications of the American Mathematical Society)

   We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]). The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172]. 

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3. On the Fractional Dynamics of Kinks in Sine-Gordon Models

   https://doi.org/10.3390/math13020220  

   Bountis, Tassos; Cantisán, Julia; Cuevas-Maraver, Jesús; Macías-Díaz, Jorge Eduardo; Kevrekidis, Panayotis G (January 2025, Mathematics)

   In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order β of the temporal derivative to that of a Caputo fractional type and found that, for 1<β<2, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order α on the spatial derivative. Here, depending on whether this order was below or above the harmonic value α=2, we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved. 

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4. Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy

   https://doi.org/10.1016/j.physd.2025.134954  

   Sharma, Harsh; Draxl-Giannoni, Juan Diego; Kramer, Boris (December 2025, Physica D: Nonlinear Phenomena)

   Existing model reduction techniques for high-dimensional models of conservative partial differential equations (PDEs) encounter computational bottlenecks when dealing with systems featuring non-polynomial nonlinearities. This work presents a nonlinear model reduction method that employs lifting variable transformations to derive structure-preserving quadratic reduced-order models for conservative PDEs with general nonlinearities. We present an energy-quadratization strategy that defines the auxiliary variable in terms of the nonlinear term in the energy expression to derive an equivalent quadratic lifted system with quadratic system energy. The proposed strategy combined with proper orthogonal decomposition model reduction yields quadratic reduced-order models that conserve the quadratized lifted energy exactly in high dimensions. We demonstrate the proposed model reduction approach on four nonlinear conservative PDEs: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, the two-dimensional Klein–Gordon equation with parametric dependence, and the two-dimensional Klein–Gordon–Zakharov equations. The numerical results show that the proposed lifting approach is competitive with the state-of-the-art structure-preserving hyper-reduction method in terms of both accuracy and computational efficiency in the online stage while providing significant computational gains in the offline stage. 

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5. On Modified Scattering for 1D Quadratic Klein–Gordon Equations With Non-Generic Potentials

   https://doi.org/10.1093/imrn/rnac010  

   Lindblad, Hans; Lührmann, Jonas; Schlag, Wilhelm; Soffer, Avy (February 2022, International Mathematics Research Notices)

   Abstract We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein–Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to continue the investigation of the occurrence of a novel modified scattering behavior of the solutions that involves a logarithmic slow-down of the decay rate along certain rays. This phenomenon is ultimately caused by the threshold resonance of the linear Klein–Gordon operator. It was previously uncovered for the special case of the zero potential in [51]. The Klein–Gordon model considered in this paper is motivated by the asymptotic stability problem for kink solutions arising in classical scalar field theories on the real line. 

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