A KdV-like advection–dispersion equation with some remarkable properties

Abstract

We discuss a new non-linear PDE, u t + (2u xx /u)u x = u xxx , invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space-time reflection-symmetric first order advection-dispersion equations. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the KdV solitary wave solution. It provides a bridge between non-linear advection, diffusion and dispersion. Special cases include the mKdV and linear dispersive equations. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, it possesses solitary and periodic travelling waves. Moreover, numerical simulations reveal recurrence properties usually associated with integrable systems. KdV and SIdV are the simplest in an infinite dimensional family of equations sharing the KdV solitary wave. SIdV and its generalizations may serve as a testing ground for numerical and analytical techniques and be a rich source for further explorations.

Solução do tipo peakon para uma equação evolutiva de terceira ordem

“…Em um recente trabalho [5], os autores procuravam equações admitindo soluções deste tipo usando algoritmos genéticos para a busca. Eles esperavam obter, como um primeiro exemplo, a própria equação KdV.…”

“…Tal equação foi chamada de equação de SIdV (scale invariant KdV). Ainda em [5], os autores deduziram outras soluções para (1). Em um trabalho recente, encontramos o grupo de simetrias da equação, veja [3].…”

Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities

“…It is well known that many well known methods have been established in scientific fields and engineering to determine the solutions with distinct physical structures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Examples of the methods that have been used are the Hirota bilinear method, the simplified Hirota's method, the Bäcklund transformation method, Darboux transformation, Pfaffian technique [7][8][9][10], the inverse scattering method, the Painlevé analysis, the generalized symmetry method, the subsidiary ordinary differential equation method, the coupled amplitude-phase formulation, the sine-cosine method, the sech-tanh method [13][14][15][16][17][18][19][20], the mapping and the deformation approach and many other methods [20][21][22][23][24][25][26][27].…”

“…The authors in [3] were conducting an investigation to improve the efficiency and accuracy of a genetic programming-based method to deduce model equations from a known analytic solution, and when using the solitary wave solution (3), the program, instead of giving the KdV equation, surprisingly gave the following KdV-like equation…”

Sobre propriedades de invariância de uma nova equação dispersiva

“…No mesmo trabalho onde foi introduzida, algumas propriedades de tal equação foram consideradas, principalmente a construção de leis de conservação. De acordo com os resultados obtidos em [4], o caso = −2/3é um caso bastante especial, principalmente naquilo que tange a obtenção de leis de conservação.Neste trabalho, estudaremos a equação (1) do ponto de vista de simetrias de Lie. De fato, mostraremos que os operadores…”