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1. Learning Interactions in Reaction Diffusion Equations by Neural Networks

   https://doi.org/10.3390/e25030489  

   Chen, Sichen; Brunel, Nicolas J-B.; Yang, Xin; Cui, Xinping (March 2023, Entropy)

   Partial differential equations are common models in biology for predicting and explaining complex behaviors. Nevertheless, deriving the equations and estimating the corresponding parameters remains challenging from data. In particular, the fine description of the interactions between species requires care for taking into account various regimes such as saturation effects. We apply a method based on neural networks to discover the underlying PDE systems, which involve fractional terms and may also contain integration terms based on observed data. Our proposed framework, called Frac-PDE-Net, adapts the PDE-Net 2.0 by adding layers that are designed to learn fractional and integration terms. The key technical challenge of this task is the identifiability issue. More precisely, one needs to identify the main terms and combine similar terms among a huge number of candidates in fractional form generated by the neural network scheme due to the division operation. In order to overcome this barrier, we set up certain assumptions according to realistic biological behavior. Additionally, we use an L2-norm based term selection criterion and the sparse regression to obtain a parsimonious model. It turns out that the method of Frac-PDE-Net is capable of recovering the main terms with accurate coefficients, allowing for effective long term prediction. We demonstrate the interest of the method on a biological PDE model proposed to study the pollen tube growth problem. 

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2. An Inverse Boundary Value Problem for a Semilinear Wave Equation on Lorentzian Manifolds

   https://doi.org/10.1093/imrn/rnab088  

   Hintz, Peter; Uhlmann, Gunther; Zhai, Jian (May 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness. 

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3. Integrable fractional modified Korteweg–deVries, sine-Gordon, and sinh-Gordon equations

   https://doi.org/10.1088/1751-8121/ac8844  

   Ablowitz, Mark J; Been, Joel B; Carr, Lincoln D (August 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to t α , α > 0, while in anomalous dispersion, the speed of localized waves is proportional to A α , where A is the amplitude of the wave. Fractional extensions of the modified Korteweg–deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion. 

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4. The scattering map determines the nonlinearity

   https://doi.org/10.1090/proc/16297  

   Killip, Rowan; Murphy, Jason; Visan, Monica (March 2023, Proceedings of the American Mathematical Society)

   Using the two-dimensional nonlinear Schrödinger equation as a model example, we present a general method for recovering the nonlinearity of a nonlinear dispersive equation from its small-data scattering behavior. We prove that under very mild assumptions on the nonlinearity, the wave operator uniquely determines the nonlinearity, as does the scattering map. Evaluating the scattering map on well-chosen initial data, we reduce the problem to an inverse convolution problem, which we solve by means of an application of the Beurling–Lax Theorem. 

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5. Fractional integrable Toda lattice and hierarchy

   https://doi.org/10.1098/rspa.2024.0812  

   Ablowitz, Mark J; Been, Joel B; Fridberg, Kyle (May 2025, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   A fractional extension of the integrable Toda lattice with decaying data on the line is obtained. Completeness of squared eigenfunctions of a linear discrete real tridiagonal eigenvalue problem is derived. This completeness relation allows nonlinear evolution equations expressed in terms of operators to be written in terms of underlying squared eigenfunctions and is related to a discretization of the continuous Schrödinger equation. The methods are discrete counterparts of continuous ones recently used to find fractional integrable extensions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger (NLS) equations. Inverse scattering transform (IST) methods are used to linearize and find explicit soliton solutions to the integrable fractional Toda (fToda) lattice equation. The methodology can also be used to find and solve fractional extensions of a Toda lattice hierarchy. 

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