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1. Reverse Space‐Time Nonlocal Sine‐Gordon/Sinh‐Gordon Equations with Nonzero Boundary Conditions

   https://doi.org/10.1111/sapm.12222  

   Ablowitz, Mark J. ; Feng, Bao‐Feng ; Luo, Xu‐Dan ; Musslimani, Ziad H. ( July 2018 , Studies in Applied Mathematics)

   Nonlocal reverse space‐time Sine/Sinh‐Gordon type equations were recently introduced. They arise from a remarkably simple nonlocal reduction of the well‐known AKNS scattering problem, hence, they constitute an integrable evolution equations. Furthermore, the inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, the IST for these novel nonlocal equations corresponding to nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complex due to the intricate branching structure of the associated linear eigenfunctions. Two cases are analyzed, which correspond to two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1‐soliton and 2‐soliton solutions are found. Both spatially independent and spatially dependent boundary conditions are considered.

    

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2. Integrable Nonlocal Nonlinear Equations

   https://doi.org/10.1111/sapm.12153  

   Ablowitz, Mark J. ; Musslimani, Ziad H. ( November 2016 , Studies in Applied Mathematics)

   A nonlocal nonlinear Schrödinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced “potential” issymmetric thus the nonlocal NLS equation is alsosymmetric. In this paper, newreverse space‐timeandreverse timenonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space‐time, and in some cases reverse time, nonlocal NLS, modified Korteweg‐deVries (mKdV), sine‐Gordon, (1 + 1) and (2 + 1) dimensional three‐wave interaction, derivative NLS, “loop soliton,” Davey–Stewartson (DS), partiallysymmetric DS and partially reverse space‐time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space‐time and reverse time nonlocal discrete nonlinear Schrödinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlevé type equations are derived from the reverse space‐time and reverse time nonlocal NLS equations.

    

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3. Soliton–mean field interaction in Korteweg–de Vries dispersive hydrodynamics

   https://doi.org/10.1111/sapm.12615  

   Ablowitz, Mark J. ; Cole, Justin T. ; El, Gennady A. ; Hoefer, Mark A. ; Luo, Xu‐Dan ( July 2023 , Studies in Applied Mathematics)

   The mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction.

    

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4. 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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5. Green's function for anisotropic dispersive poroelastic media based on the Radon transform and eigenvector diagonalization

   https://doi.org/10.1098/rspa.2018.0610  

   Zhan, Qiwei ; Zhuang, Mingwei ; Fang, Yuan ; Liu, Jian-Guo ; Liu, Qing Huo ( January 2019 , Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   A compact Green's function for general dispersive anisotropic poroelastic media in a full-frequency regime is presented for the first time. First, starting in a frequency domain, the anisotropic dispersion is exactly incorporated into the constitutive relationship, thus avoiding fractional derivatives in a time domain. Then, based on the Radon transform, the original three-dimensional differential equation is effectively reduced to a one-dimensional system in space. Furthermore, inspired by the strategy adopted in the characteristic analysis of hyperbolic equations, the eigenvector diagonalization method is applied to decouple the one-dimensional vector problem into several independent scalar equations. Consequently, the fundamental solutions are easily obtained. A further derivation shows that Green's function can be decomposed into circumferential and spherical integrals, corresponding to static and transient responses, respectively. The procedures shown in this study are also compatible with other pertinent multi-physics coupling problems, such as piezoelectric, magneto-electro-elastic and thermo-elastic materials. Finally, the verifications and validations with existing analytical solutions and numerical solvers corroborate the correctness of the proposed Green's function. 

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