Partial differential systems with non-local nonlinearities: generation and solutions

@article{Beck2018PartialDS,
  title={Partial differential systems with non-local nonlinearities: generation and solutions},
  author={Margaret Beck and Anastasia Doikou and Simon J. A. Malham and Ioannis Stylianidis},
  journal={Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  year={2018},
  volume={376}
}
  • M. Beck, A. Doikou, Ioannis Stylianidis
  • Published 26 September 2017
  • Mathematics
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples, including reaction… 

Figures from this paper

Grassmannian flows and applications to nonlinear partial differential equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial

Applications of Grassmannian flows to integrable systems

We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable in the sense that they are realisable as Fredholm Grassmannian flows. In other words,

Integrability of local and nonlocal non-commutative fourth order quintic nonlinear Schrodinger equations

Abstract. We prove integrability of a generalised non-commutative fourth order quintic nonlinear Schrödinger equation. The proof is relatively succinct and rooted in the linearisation method

Applications of Grassmannian and graph flows to nonlinear systems

We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable. By this we mean that solutions can be generated by solving a corresponding linear

Integrability of local and nonlocal non-commutative fourth order quintic NLS equations

We prove integrability of a generalised non-commutative fourth order quintic NLS equation. The proof is relatively succinct and rooted in the linearisation method pioneered by Ch. Poppe. It is based

Stability of nonlinear waves and patterns and related topics

TLDR
A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties, including the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves.

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture

Applications of Grassmannian and graph flows to coagulation systems

We demonstrate how many classes of Smoluchowski-type coagulation models can be realised as Grassmannian or nonlinear graph flows and are therefore linearisable, and/or integrable in this sense. We

References

SHOWING 1-10 OF 40 REFERENCES

Grassmannian flows and applications to nonlinear partial differential equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial

Computing Stability of Multidimensional Traveling Waves

TLDR
A numerical method for computing the pure-point spectrum associated with the linear stability of multidimensional traveling fronts to parabolic nonlinear systems and studies the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system.

Darboux Transformations and Solitons

In 1882 Darboux proposed a systematic algebraic approach to the solution of the linear Sturm-Liouville problem. In this book, the authors develop Darboux's idea to solve linear and nonlinear partial

The Korteweg–deVries Equation: A Survey of Results

The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma

Linear spectral problems, non-linear equations and the ∂-method

It is known that a number of non-linear partial differential equations and systems can be linearised, in principle, by solving an inverse scattering problem for an associated linear equation or

Evans function and Fredholm determinants

  • Issa KarambalS. J. Malham
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2015
TLDR
These new results include clarification of the sense in which the Evans function and transmission coefficient are equivalent and proof of the equivalence of the transmission coefficient and Fredholm determinant, in particular in the case of distinct far fields.

Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model

An integro-differential reaction-diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations

Grassmannian spectral shooting

TLDR
A new numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures and avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves.

Geometric numerical schemes for the KdV equation

TLDR
It is shown that geometrical schemes are as much robust and accurate as Fourier-type pseudospectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.