# 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} }

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…

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## References

SHOWING 1-10 OF 40 REFERENCES

### Grassmannian flows and applications to nonlinear partial differential equations

- Mathematics
- 2016

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

- Mathematics, Computer ScienceSIAM J. Appl. Dyn. Syst.
- 2009

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

- Mathematics
- 1992

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

- Physics, Mathematics
- 1976

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

- Mathematics
- 1989

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

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2015

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

- Mathematics
- 1990

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

- MathematicsMath. Comput.
- 2010

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

- Computer Science, Mathematics
- 2013

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.