Abstract
We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2.
Highlights
The Johnson equation was introduced in 1980 by Johnson [1] to describe waves surfaces in shallow incompressible fluids [2, 3]
We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N + 1) in x, t and 4N(N + 1) in y depending on 2N − 2 parameters
We have constructed solutions to the Johnson equation, starting from the solutions of the KPI equation, what makes it possible to obtain rational solutions. These solutions are expressed by means of quotients of two polynomials of degree 2N(N + 1) in x, t and 4N(N + 1) in y depending on 2N − 2 parameters
Summary
The Johnson equation was introduced in 1980 by Johnson [1] to describe waves surfaces in shallow incompressible fluids [2, 3]. The KPI equation first appeared in 1970 [7] in a paper written by Kadomtsev and Petviashvili This equation is considered as a model for surface and internal water waves [10] and in nonlinear optics [11]. We will use the KPI equation to construct solutions to the Johnson equation but in another way different from this used in [6]. These last authors consider another representation of KPI equation given by (ut + 6uux + uxxx)x − 3uyy = 0,. We construct only rational solutions of order 3, depending on 4 real parameters; we construct the representations of their modulus in the plane of the coordinates (x, y) according to the four real parameters ai and bi for 1 ≤ i ≤ 2 and time t
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
