# Moutard transform for the conductivity equation.

@article{Grinevich2017MoutardTF, title={Moutard transform for the conductivity equation.}, author={P. G. Grinevich and R. G. Novikov L.D. Landau Institute for Theoretical Physics and Ras and Russia. and Lomonosov Moscow State University and Cmap and Ecole Nationale Polytechnique and France. and Institute of Earthquake Prediction}, journal={arXiv: Mathematical Physics}, year={2017} }

We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions. In addition, at least, some of the Darboux-Moutard type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrodinger equation at zero energy are also shown.

## One Citation

### Darboux Moutard Transformations and Poincaré—Steklov Operators

- MathematicsProceedings of the Steklov Institute of Mathematics
- 2018

Formulas relating Poincaré–Steklov operators for Schrödinger equations related by Darboux–Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the…

## References

SHOWING 1-10 OF 22 REFERENCES

### The Moutard transformation of two-dimensional Dirac operators and Möbius geometry

- Mathematics
- 2014

We describe the action of inversion on given Weierstrass representations for surfaces and show that the Moutard transformation of two-dimensional Dirac operators maps the potential (the Weierstrass…

### The Moutard transformation of two-dimensional Dirac operators and the conformal geometry of surfaces in four-dimensional space

- Mathematics, Physics
- 2016

The Moutard transformation for the two-dimensional Dirac operator with complexvalued potential is constructed. It is shown that this transformation binds the potentials of Weierstrass representations…

### Generalized analytic functions, Moutard-type transforms, and holomorphic maps

- Mathematics
- 2015

We continue the study of a Moutard-type transform for generalized analytic functions, which was initiated in [1]. In particular, we suggest an interpretation of generalized analytic functions as…

### Darboux transformation for the modified Veselov-Novikov equation

- Mathematics
- 2002

A Darboux transformation is constructed for the modified Veselov-Novikov equation. By means of the Darboux transformation, two families of explicit solutions of this equation are given.

### 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…

### Darboux Moutard Transformations and Poincaré—Steklov Operators

- MathematicsProceedings of the Steklov Institute of Mathematics
- 2018

Formulas relating Poincaré–Steklov operators for Schrödinger equations related by Darboux–Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the…

### Blowing up solutions of the modified Novikov-Veselov equation and minimal surfaces

- Mathematics
- 2014

We propose a construction of blowup solutions of the modified Novikov-Veselov equation based on the Moutard transformation of the two-dimensional Dirac operators and on its geometric interpretation…

### A Boundary Value Problem for Conjugate Conductivity Equations

- Mathematics
- 2015

We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half‐plane with conductivity σ(x,y)=xp , p∈Z* . The representations…

### Moutard type transformation for matrix generalized analytic functions and gauge transformations

- Physics
- 2016

A Moutard type transformation for matrix generalized analytic functions is derived. Relations between Moutard type transforms and gauge transformations are demonstrated.