Computing Travelling Wave Solutions (TWS) for non-linear PDE systems

Edgardo Cheb-Terrab, MITACS-CECM, SFU

    Given a non-linear PDE system in unknowns f[i](x[j]), a travelling
    wave solution is an exact closed form solution of the form

                                    n[i]
                                    -----
                                     \               k
                        f[i](tau) =   )   A[i, k] tau
                                     /
                                    -----
                                    k = 1

    where the n[i] are finite, the A[i,k] are constants with respect to
    the independent variables x[j], and

                                      j
                                    -----
                                     \
                         tau = tanh(  )   C[k] x[k])
                                     /
                                    -----
                                    k = 1

    where the C[k] are constants with respect to the x[j]. This type of
    solution plays an important role in the study of non-linear physical
    phenomena (fluid dynamics, elastic media, field theories etc.). In
    this talk the way TWS are constructed is reviewed (includes a demo of
    a Maple implementation), the generality of these solutions is analyzed
    and a generalization of the method taking tau as the solution of an
    arbitrary Riccati equation is discussed.