On the Existence of Blow-up Phenomenon of Generalized ( 2 + 1 ) Dimensional Kaup — Kupershmidt Equation

In this paper, We study and prove the existence of blow-up solution of generalized (2+1) dimensional Kaup—Kupershmidt (KK) equation by direct construction of an exact blow-up solution . Since Sawada-Kotera equation and Kaup-Kupershmidt (K. Sawada, 1974) equation are only particular case of our generalized equation , we can also prove the existence of blowup solution of Sawada-Kotera equation and Kaup-Kupershmidt (K. Sawada, 1974) equation by our construction of exact solution.

The extent of applicability of this method crucially depends on the dimension of space.There are a great number of various nonlinear evolution equations in 1 + 1 dimensions (one spatial and one time dimension) integrable by the inverse scattering transform method (V.E.Zakharov, 1980; M.J. Ablowitz, 1981;F. Calogero, 1982).For two spatial dimensions the class of integrable equations is much poorer.The best known two-dimensional integrable equations which have physical applications are the Kadomtsev-Petviashvili equation, three-wave equations and the Davey-Stewarson equation (V.E.Zakharov, 1980;M.J. Ablowitz, 1981).
In the present paper we consider a nonlinear evolution equation in two spatial dimensions (x, y) connected with the differential operator where ∂x ≡ ∂/∂x, ∂y ≡ ∂/∂y, V0 (x, y, t), . . ., VN (x, y, t) are scalar functions.The applicability of the inverse scattering transform method to the problem has been demonstrated in ref. (V.E. Zakharov, 1974).
We present here some nonlinear evolution equations in 2 + 1 dimensions (x, y, t) for one dependent variable u(x,y, t) which can be represented as the commutativity condition [L, T] = LT-TL = 0.The operator L is of the form (1). The operators T will be given explicitly.The obtained equations are the two dimensional generalizations of the well-known Gardner (combined KdV and modified KdV equation ) equation, the Sawada-Kotera, the Kaup-Kupershmidt and the Harry Dim equations.Now we consider the operator L of the third order.The first example is The operators ( 1) and ( 2) commute if (3) is the two-dimensional integrable generalization of the Sawada-Kotera equation (K.Sawada, 1974).
Another example is www.ccsenet.org/jmr The corresponding evolution equation is (6)is the equation which was considered by Kaup and Kupershmidt (D.J. Kaup, 1980).
Let us emphasize that the parts of the two-dimensional Sawada-Kotera and Kaup-Kupershmidt equations ( 3) and ( 4) which contain the terms with derivative ∂ y coincide.It is also easily seen that eqs.
All equations considered have infinite series of the integrals of motion.They can be obtained by a standard recursion procedure if one represents the solution Ψ of the equation L Ψ = 0 in the form (see refs.V.E.Zakharov, 1980; M.J. Ablowitz , 1981;D.J. Kaup, 1980) Similar to their one-dimensional limit the two-dimensional equations ( 3), ( 4), ( 7), ( 9), ( 11) may have various physical applications.

Main Result
Now, we can introduce the (2+1) dimensional generalized Kaup-Kupershmidt equation (B.Konopelchenko, 1984; M.J. Ablowitz, 1991;S.B. Leble, 1994): Where a, b, c, d, e, f, g, h, i are all nonzero constant.Sawada-Kotera equation and Kaup-Kupershmidt equation are only special form of the equation ( 11) In (Xing-Biao Hu, 1999), A bilinear form for the equation is found and then 3-soliton solutions are obtained with the assistance of software Mathematica.Six symmetries of the bilinear 2+1 dimensional KK equation are given and their symmetry algebra is identified.But up to now, no one consider the general case (11).
(11) is a nonlinear, fifth order, 3-dimentional, differential integral equation.It is extremely difficult.To our knowledge, very little is known for this equation.However, we still successfully construct an analytic exact solution for it.Our solution is a blowing up solution.The solution will blow up in the neighborhood of the plane px + qy x is an partial integration operator, and We assume the KK (12) equation has the blowing up solution of the form: Where A, B, p, q, r, s, are the undetermined coefficient 12) into (11), then Equating both sides, we get Now, we have only 3 equations as restrictions, but we have 5 free variables (A, B, p, q, r), so there are infinitely many solutions.Now we regard p, q as free parameters, A, B, r as variables, to solve (13).
Where p, q are free parameters.

If e=0,
In the above solution of algebra equation.We firstly express A in terms of free parameter and coefficient, then express B in terms of A and free parameter and coefficient, and express r in terms of B and free parameter and coefficient, in this way we express A, B, r in terms of free parameter and coefficient.What's more, at first, we assume all coefficient are non-zero, but in the process we find that we can still manage to find the solution even if e is zero, therefore, the requirement of keeping e to be non-zero constant can be removed.And we should notice that the solution is complex when e 0 ( 16) and (12cp 2 + 6p 2 d 2 ) − 1440bp 4 < 0 (17) and the solution is real when e=0 www.ccsenet.org/jmr