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In this work, symmetry groups are used to determine symmetry reductions of a (2 + 1)‐dimensional Keller–Segel system depending on two arbitrary functions. We show that the point symmetries of the considered Keller–Segel system comprise an infinite‐dimensional Lie algebra which involves three arbitrary functions. By way of example, we have used these point symmetries to reduce straightaway the given system of second‐order partial differential equations to a system of second‐order ordinary differential equations. Moreover, we are allowed to substitute one of the dependent variables from one of the equations into the other, leading to an equivalent fourth‐order nonlinear ordinary differential equation. This equation is reduced through the use of solvable symmetry subalgebras, and some exact solutions are obtained for a particular case. [ABSTRACT FROM AUTHOR]

A generalization of the time-delayed Burgers–Fisher equation is studied. This partial differential equation appears in many physical and biological problems describing the interaction between reaction, diffusion, and convection. New travelling wave solutions are obtained. The solutions are derived in a systematic way by applying the multi-reduction method to the symmetry-invariant conservation laws. The translation-invariant conservation law yields a first integral, which is a first-order Chini equation. Under certain conditions on the coefficients of the equation, the Chini type equation obtained can be solved, yielding travelling wave solutions expressed in terms of the Lerch transcendent function. For a special case, the first integral becomes a Riccati equation, whose solutions are given in terms of Bessel functions, and for a special case of the parameters, the solutions are given in terms of exponential, trigonometric, and hyperbolic functions. Furthermore, a complete classification of the zeroth-order local conservation laws is obtained. To the best of our knowledge, our results include new solutions that have not been previously reported in the literature. [ABSTRACT FROM AUTHOR]

In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling waves solutions, all the the conservation laws that are invariant under translation symmetries are directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution equations under consideration into a first-order autonomous equation. [ABSTRACT FROM AUTHOR]

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs. [ABSTRACT FROM AUTHOR]

Background: Our study aims to comment on all ADPKD variants identified in our health area and explain how they are distributed geographically, to identify new variants, and relate the more frequent variants with their renal phenotype in terms of kidney survival. Materials and Methods: We identified patients suffering from ADPKD in a specialized consultation unit; genealogical trees were constructed from the proband. According to the ultrasound-defined modified Ravine–Pei criteria, relatives classified as at risk were offered participation in the genetic study. Socio-demographic, clinical, and genetic factors related to the impact of the variant on the survival of the kidney and the patient, such as age at RRT beginning and age of death, were recorded. Results: In 37 families, 33 new variants of the PKD1 gene were identified, which probably produce a truncated protein. These variants included 2 large deletions, 19 frameshifts, and 12 stop-codons, all of which had not been previously described in the databases. In 10 families, six new probably pathogenic variants in the PKD2 gene were identified. These included three substitutions; two deletions, one of which was intronic and not associated with any family; and one duplication. A total of 11 missense variants in the PKD1 gene were grouped in 14 families and classified as probably pathogenic. We found that 33 VUS were grouped into 18 families and were not described in the databases, while another 15 were without grouping, and there was only 1 in the PKD2 gene. Some of these variants were present in patients with a different pathogenic variant (described or not), and the variant was probably benign. Renal survival curves were compared to nonsense versus missense variants on the PKD1 gene to check if there were any differences. A group of 328 patients with a nonsense variant was compared with a group of 264 with a missense variant; mean renal survival for truncated variants was lower (53.1 ± 0.46 years versus non-truncated variant 59.1 ± 1.36 years; Log Rank, Breslow, and Tarone Ware, p < 0.05). Conclusions: To learn more about ADPKD, it is necessary to understand genetics. By describing new genetic variants, we are approaching creation of an accurate genetic map of the disease in our country, which could have prognostic and therapeutic implications in the future. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fourth‐order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one‐dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one‐dimensional subalgebras. Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves. [ABSTRACT FROM AUTHOR]

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G (u) . We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G ′ (u) ≠ 0 , we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov's method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations. [ABSTRACT FROM AUTHOR]