Source Generation Procedure Research Articles

A Modified Semi-Discrete BKP-Type Equation with Self-Consistent Sources

In this paper, a modified semi-discrete BKP-type equation [ t -1 -flow of the BKP hierarchy (KP hierarchy of the B-type)] is firstly proposed and its pfaffian solutions are obtained. Then, utilizing the source generation procedure, this modified semi-discrete BKP-type equation with self-consistent sources (BKP-type ESCS) is presented and its pfaffian solutions are derived. This modified semi-discrete BKP-type ESCS obtained is shown to constitute a bilinear Backlund transformation for the semi-discrete BKP-type ESCS originally given by Wang et al. in 2007. This means that the commutativity of the source generation procedure (SGP) and Backlund transformations (BT) is valid for the semi-discrete BKP-type equation.

BÄCKLUND TRANSFORMATION OF A NON-ISOSPECTRAL KPESCS AND ITS NONLINEAR COUPLED SYSTEM

A non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources (KPESCS) is first derived by using the source generation procedure. Moreover, its bilinear Bäcklund transformation is obtained. Furthermore, a new nonlinear coupled system of the non-isospectral KPESCS and its exact solutions are established. The new coupled system cannot only be reduced to the non-isospectral KPESCS but can also be reduced to the coupled non-isospectral KP equation.

The non-isospectral modified Kadomtsev–Petviashvili equation with self-consistent sources and its coupled system

A new algebraic method called ‘source generation procedure’ is applied to construct non-isospectral soliton equations with self-consistent sources. As results, the non-isospectral modified Kadomtsev–Petviashvili equation with self-consistent sources (mKPESCS) and its Gramm-type determinant solutions are obtained by using the source generation procedure. Furthermore, a new coupled system of the non-isospectral mKPESCS and its Pfaffian solutions are constructed.

Commutativity of source generation procedure and Bäcklund transformation: A BKP equation

The source generation procedure is applied to a BKP-type equation and a modified BKP-type (m-BKP) equation, respectively. As a result, a BKP equation with self-consistent sources (ESCS) and a m-BKP ESCS are derived. It is also proved that the bilinear m-BKP ESCS constitutes a bilinear Bäcklund transformation for the BKP ESCS. This means that the commutativity between the source generation procedure and Bäcklund transformation is valid for the BKP equation.

Ishimori-I equation with self-consistent sources

In this paper, Grammian solutions of the Ishimori-I (Ish-I) equation are first obtained by Hirota’s direct method. Utilizing the source generation procedure, this equation with self-consistent sources is then presented and the corresponding Grammian solutions are derived. Finally, as a simple case, the (1,1) dromion solution is examined.

Lie symmetries of soliton equations with self-consistent sources via source generation procedure

In this paper, the ‘source generation’ procedure (SGP) proposed by Hu and Wang [X.B. Hu, H.Y. Wang, Construction of dKP and BKP equation with self-consistent sources, Inverse Problems 22 (2006) 1903–1920] is utilized to derive Lie symmetries of bilinear soliton equations with self-consistent sources (SESCS) such as KPESCS, BKPESCS, and differential–difference KPESCS. Furthermore, it is shown that these Lie symmetries constitute generators of the corresponding Lie symmetry algebras.

The Nizhnik-Veselov-Novikov equation with self-consistent sources

We construct the coupled generalization of the Nizhnik-Veselov-Novikov (NVN) equation, i.e., the NVN equation with self-consistent sources (ESCS) and solve it using the so-called source generation procedure. We obtain BKP-type and DKP-type Pfaffian solutions of the NVN ESCS. In addition, we give a bilinear Backlund transformation and a Lax pair for the NVN ESCS. Furthermore, this Lax pair can be reduced to a Lax pair for the NVN equation.

Source generation of the Davey-Stewartson equation

The “source generation” procedure (SGP) proposed by Hu and Wang [Inverse Probl. 22, 1903 (2006)] provides a new way to systematically generate so-called soliton equations with self-consistent sources. In this paper, we apply this SGP to a Davey-Stewartson (DS) equation based on the Hirota bilinear form, producing a system of equations which is called the DS equation with self-consistent sources (DSESCS). Meanwhile, we obtain the Gramm-type determinant solutions to the DSESCS. Since the DS equation is a (2+1)-dimensional integrable generalization of the nonlinear Schrödinger (NLS) equation, the DSESCS may be viewed as a (2+1)-dimensional integrable generalization of the nonlinear Schrödinger equation with self-consistent sources. These results indicate the commutativity of source generation procedure and (2+1)-dimensional integrable generalizations for the NLS equation.

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

The pfaffian solution of a semi-discrete BKP-type equation is obtained at first, then utilizing the source generation procedure, this equation with self-consistent sources (BKPESCS) is presented and its pfaffian solutions are derived. Finally, a bilinear Bäcklund transformation for the semi-discrete BKPESCS is given.

New type of Kadomtsev–Petviashvili equation with self-consistent sources and its bilinear Bäcklund transformation

A new type of the KP equation with self-consistent sources (KPESCS) first found by Mel'nikov (1983 Lett. Math. Phys. 7 129–36) is re-constructed via source generation procedure. A new feature of the obtained KPESCS is that we allow y-dependence of the arbitrary constants in the determinantal solution for the KP equation while applying the source generation procedure. We also propose a new idea of commutativity of source generation procedure and Bäcklund transformations to generate a BT for the new KPESCS which indicates the integrability of the KPESCS.