Recent work has developed a broad generalization of the double-reduction method by considering the space of invariant conservation laws with respect to a given symmetry. In its simplest formulation, the generalization is able to reduce a nonlinear PDE in $2$ variables to an ODE with $m$ first integrals where $m$ is the dimension of the space of invariant conservation laws. Nonlinear PDEs in $3$ or more variables can be reduced to an ODE similarly by using an algebra of given symmetries. Importantly, the algebra does not need to be solvable.

The general method employs multipliers and is fully algorithmic. In particular, no a priori knowledge of conservation laws of the nonlinear PDE is necessary, and the multi-reduction is carried out in one step.

In this talk, a summary of the general multi-reduction method will be presented for obtaining invariant solutions of physically interesting PDEs. Examples will be shown for quadruple reduction from a single symmetry; complete integration from a solvable algebra in one step; reduction via a non-solvable algebra.

Cellular adhesion is one of the most important interaction forces between cells and other tissue components. In 2006, Armstrong, Painter and Sherratt introduced a non-local PDE model for cellular adhesion, which was able to describe known experimental results on cell sorting and cancer growth. The analysis becomes challenging through non-local cell-cell interaction and interactions with boundaries. In this talk I will use symmetry methods to analyse aggregations and pattern formation of the non-local adhesion model. (joint work with A. Buttenschoen).

We consider $u(t,x)$ as an octonion variable in evolution equations $u_t = F(u,u_x,u_{xx},u_{xxx})$, and we aim to find a Lax pair $L_t = [M,L]$ where $L$ and $M$ are linear differential operators in terms of $\partial_x$ with coefficients involving $u$ and $x$-derivatives of $u$. For $F$, we assume it is homogeneous under a scaling of $t,x,u$ which is either the scaling in the KdV equation or the mKdV equation. This gives a polynomial ansatz with undetermined (real) coefficients. Similarly, for $L$ and $M$, we assume they are scaling homogeneous, where the scaling weight of $M$ is the same as that of $\partial_t$ while the scaling weight of $L$ can be chosen freely.

The determining condition is $\big(L_t -[M,L]\big)\big|_{u_t = F} =0$. We split this condition in the jet space of $u$, and do a further splitting with respect to a real basis (8-dimensional) for the octonions. This gives a large overdetermined system in the undetermined (real) coefficients in ansatz for $F$, $L$, $M$. We use Maple to do the splittings, and depending on the complexity of the system, we solve it using 'rifsimp' in Maple or a package called 'Crack' in Reduce.

As a main result, we obtain a single KdV octonion equation, three mKdV octonion equations, and also a single potential-KdV octonion equation, each of which has more than one Lax pair.