# American Institute of Mathematical Sciences

2015, 2015(special): 981-989. doi: 10.3934/proc.2015.0981

## Symmetries and solutions of a third order equation

 1 Instituto de Matemática, Estatística e Computação Cie ntífica, IMECC - UNICAMP, Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil 2 Centro de Matemática, Computação e Cognição, Universidade Federal do ABC - UFABC, Rua Santa Adélia, 166, Bairro Bangu, 09.210 -- 170, Santo André, SP

Received  September 2014 Revised  January 2015 Published  November 2015

In this paper we study a new third order evolution equation discovered a couple of years ago using a genetic programming. We show that the Lie symmetries of this equation corresponds to space and time translations, as well as a dilation on the space of independent variables and another one with respect to the depend variable. From its symmetries, explicit solutions of the equation are obtained, some of them expressed in terms of the solutions of the Airy equation and Abel equation of the second kind. Additionally, by using the direct method we establish three conservation laws for the equation, one of them new.
Citation: Júlio Cesar Santos Sampaio, Igor Leite Freire. Symmetries and solutions of a third order equation. Conference Publications, 2015, 2015 (special) : 981-989. doi: 10.3934/proc.2015.0981
##### References:

show all references

##### References:
 [1] Mathew Gluck. Classification of solutions to a system of $n^{\rm th}$ order equations on $\mathbb R^n$. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 [2] Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307 [3] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377 [4] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [5] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [6] Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 [7] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [8] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [9] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [10] Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002 [11] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [12] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [13] Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $p$-Laplacian equations on $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265 [14] Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $\mathcal{W}(a, b, r)$. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123 [15] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [16] Jiahao Qiu, Jianjie Zhao. Maximal factors of order $d$ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278 [17] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [18] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [19] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [20] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

Impact Factor: