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Pfaffian solution of a semi-discrete BKP-type equation and its source generation version

“…Therefore τ n in ( 9) is a pfaffian solution of equation (8). Here the function τ n is only a formal solution, and for application of the pf(i, j ) n in (10), we can take…”

“…In this part, we will apply the source generation procedure to the bilinear equation (8). We first change the function τ n in (9) into the form…”

“…where each C i (y) is an arbitrary function of the variable y. Then the function f n will no longer satisfy equation (8). Following the source generation procedure, we introduce new functions expressed by…”

“…Hence it has become an interesting problem to study how to propose and solve SESCSs. There exist several ways to study SESCSs, such as the inverse scattering transform, the Darboux transformation and Hirota's method (see [5][6][7][8][9][10][11][12][13][14][15]). Recently, a new source generation procedure [16] has been found to construct and solve SESCSs based on the bilinear method.…”

Toda lattice with a special self-consistent source

“…We note the relevant works [7], [8], where Korteweg-de Vries equations with a self-consistent source were considered in relation to applying the inverse scattering method for a Sturm-Liouville operator.…”