Exact Anti-Self-Dual four-manifolds with a Killing symmetry by similarity transformations

We study the group properties and the similarity solutions for the constraint conditions of anti-self-dual null Kähler four-dimensional manifolds with at least a Killing symmetry vector. Specifically we apply the theory of Lie symmetries to determine all the infinitesimal generators of the one-parameter point transformations which leave the system invariant. We use these transformations to define invariant similarity transformations which are used to simplify the differential equations and find the exact form of the spacetime. We show that the constraint equations admit an infinite number of symmetries which can be used to construct an infinite number of similarity transformations.

[2]  M. Pandey.  Lie Symmetries and Exact Solutions of Shallow Water Equations with Variable Bottom , 2015 .

[3]  R. Arora,et al.  Invariance analysis, optimal system and conservation laws of $$(2+1)$$ ( 2 + 1 ) -dimensional non-linear Vakhnenko equation , 2021, Pramana.

[4]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[5]  Hanze Liu,et al.  Nonlocal symmetries and exact solutions of the (2+1)-dimensional generalized variable coefficient shallow water wave equation , 2019, Appl. Math. Lett..

[6]  R. Popovych,et al.  Lie symmetry analysis and exact solutions of the quasigeostrophic two-layer problem , 2010, 1010.1542.

[7]  Decio Levi,et al.  Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra , 1986 .

[8]  R. Arora,et al.  Lie symmetry analysis, optimal system and invariant solutions of (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles , 2021, The European Physical Journal Plus.

[9]  R. Arora,et al.  Lie symmetry analysis, optimal system, and generalized group invariant solutions of the (2 + 1)‐dimensional Date–Jimbo–Kashiwara–Miwa equation , 2020, Mathematical Methods in the Applied Sciences.

[10]  김용진,et al.  34 , 1986, Magical Realism for Non-Believers.

[11]  G. Webb.  Lie symmetries of a coupled nonlinear burgers-heat equation system , 1990 .

[12]  Yong Chen,et al.  A direct algorithm of one-dimensional optimal system for the group invariant solutions , 2014, 1411.3797.

[13]  A. Aksenov,et al.  CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 2. Applications in Engineering and Physical Sciences , 1995 .

[14]  Ben Muatjetjeja,et al.  Exact solutions of the generalized Lane–Emden equations of the first and second kind , 2011 .

[15]  S. Maharaj,et al.  Radiating stars with exponential Lie symmetries , 2016, 1612.08517.

[16]  Symmetry and conservation law structures of some anti-self-dual (ASD) manifolds , 2016 .

[17]  A. Chesnokov.  Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow , 2008 .

[18]  C. LeBrun.  Explicit self-dual metrics on $\mathbb{CP}_2 \# \cdots\#\mathbb{CP}_2$ , 1991 .

[19]  B. M. Fulk.  MATH , 1992 .

[20]  S. Maharaj,et al.  Charged radiating stars with Lie symmetries , 2019, The European Physical Journal C.

[21]  J. Meinhardt,et al.  Symmetries and differential equations , 1981 .

[22]  S. Hawking,et al.  Gravitational Multi - Instantons , 1978 .

[23]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[24]  Andrew G. Glen,et al.  APPL , 2001 .

[25]  M. Inç,et al.  Invariant and simulation analysis to the time fractional Abrahams–Tsuneto reaction diffusion system , 2019, Physica Scripta.

[26]  R. Arora,et al.  Invariance analysis, exact solutions and conservation laws of (2+1)-dimensional dispersive long wave equations , 2020 .

[27]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[28]  Michael Tsamparlis,et al.  Symmetries of Differential Equations in Cosmology , 2018, Symmetry.

[29]  Andronikos Paliathanasis.  One-Dimensional Optimal System for 2D Rotating Ideal Gas , 2019, Symmetry.

[30]  Maciej Dunajski.  Anti-self-dual four–manifolds with a parallel real spinor , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.