Conservation laws and exact solutions of the $(3+1)$-dimensional Jimbo–Miwa equation

In this article, by using the Herman–Pole technique the conservation laws of the $(3+1)-$ ( 3 + 1 ) − Jimbo–Miwa equation are obtained, and then by using the Lie symmetry analysis all of the geometric vector fields of this equation are given. Also, the non-classical symmetries of the Jimbo–Miwa equation have been determined by applying nonclassical schemes. Eventually, the ansatz solutions of the Jimbo–Miwa equations utilizing the tanh technique have been offered.

Conservation laws, classical symmetries and exact solutions of a (1 + 1)-dimensional fifth-order integrable equation

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach.

Non-classical symmetry and analytic self-similar solutions for a non-homogenous time-fractional vector NLS system

The complex PDEs are a very important and interesting task in nonlinear quantum science. Although there have been extensive studies on the classical complex models, solving the fractional complex models still has a lot of shortcomings, especially for the non-homogenous ones. Therefore, the present study focuses on solving the two-component non-homogenous time-fractional NLS system, our method is to solve a prolonged fractional system derived from the governed model. We first establish non-classical symmetries of this new enlarged system by using the fractional Lie group method. Then, with the help of fractional Erdélyi–Kober operator, we reduce this new system into fractional ODEs, the self-similar solutions are obtained via the power series expansion. The convergence of these solutions are proven as all the variable coefficients are analytic. Finally, we generalize our methods to handle the multi-component case. We conclude that this way may also bring some convenience for solving other complex systems.

On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation

The nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. Through this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. The challenge with the nonclassical symmetries method, however, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and therefore difficult to solve. In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. This is what we illustrate in this paper. We construct four nontrivial nonclassical symmetries of the Black–Scholes equation by transforming nonclassical symmetries of the heat equation. For completeness, we also construct invariant solutions of the Black–Scholes equation associated with the determined nonclassical symmetries.

Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

Q-conditional (non-classical) symmetries of the known three-component reaction-diffusion (RD) system [K. Aoki et al. Theor. Popul. Biol. 50, 1–17 (1996)] modelling interaction between farmers and hunter-gatherers are constructed for the first time. A wide variety of Q-conditional symmetries are found, and it is shown that these symmetries are not equivalent to the Lie symmetries. Some operators of Q-conditional (non-classical) symmetry are applied for finding exact solutions of the RD system in question. Properties of the exact solutions (in particular, their asymptotic behaviour) are identified and possible biological interpretation is discussed.

Quantum anomalies: A few physics applications

Chapter 17 exhibits various examples where classical symmetries cannot be transferred to quantum theories. The obstructions are characterized by anomalies. The examples involve chiral symmetries combined with currents or gauge symmetries, leading to chiral anomalies. In particular, anomalies lead to obstruction in the construction of theories. In particular, the structure of the Standard Model of particle physics is constrained by the requirement of anomaly cancellation. Other applications, like the relation between electromagnetic pi0 decay and the axial anomaly, are described. Anomalies are related to the Dirac operator index, leading to relations between anomaly and topology. To prove anomaly cancellation beyond perturbation theory, one can use lattice regularization. However, the definition of lattice chiral transformations is non–trivial. It is based on solutions of the Ginsparg–Wilson relation.