Exact solutions of the generalized $K(m,m)$ equations

Seeking to understand the role of nonlinear dispersion in the formation of patterns in liquid drops in 1993 Rosenau and Hyman [1] introduced a family of fully nonlinear K(m, n) equations and also presented solutions of the K(2, 2) equation to illustrate the remarkable behavior of these equations. The K(m, n) equations have the property that for certain m and n their solitary wave solutions have compact support. That is, they vanish identically outside a finite core region. These properties have a wide application in the fields of Physics and Mathematics, such as Nonlinear Optics, Geophysics, Fluid Dynamics and others. Later, this equation was studied by various scientists worldwide [2][3][4][5][6][7][8][9][10][11][12][13][14].

In this paper we construct periodic wave solutions for the following family of nonlinear partial differential equations

The equation ( 1) is of order 2N + 1 and depends on N + 2 parameters denoted by α 0 , …, α N , m. This family contains a number of well-known generalizations of partial differential equations which were considered before [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].

This paper is organized as follows. In Section 2 we describe a method which enables one to construct periodic wave solutions for the concerned family of nonlinear partial differential equations. In Sections 3-6 we give several specific examples for some meanings of N.

Applying traveling-wave variable

to Eq.( 1) and integrating the results yield the following Nth-order equation

The constant of integration is set to be zero. Substituting

we have p = 2 N m-1 . Note that Eq.( 3) is an autonomous equation, and we can substitute z to (zz 0 ). We will take this fact into account in final solution, but we omit this substitution in our calculations. We search solutions of Eq.(3) in the form

There is a remarkable property of a function cos(B N z). First of all we have to show expansion terms of Eq.(3).

In the case k = 1 we have the following expression

In the case k = 2 we obtain

In the case k = 3 we get

In the general case k = N derivative takes the form

where M 2N 1 , …, M 2N N are polynomials of 2N power.

Substituting Eq.( 8) into Eq.( 3) we obtain the expression

Equating coefficients at powers of cos (B N z) to zero yields an algebraic system. Solving this system we obtain the values of parameters A N , B N and correlations on the coefficients α 0 , …, α N .

The first member of the family (1) in the case N = 1 takes the form

Taking the traveling wave ansatz (2) into account, we have the equation with respect to y(z)

Following the procedure suggested in the previous section we obtain the equation

Equating coefficients at powers of cos (B 1 z) to zero we get values of parameters A 1 , B 1

Solutions of Eq.( 12) are the following

In the case α 0 = 1, α 1 = 1 formula ( 15) is a solution of a well-known

4 Periodic wave solutions of Eq.( 3) in the case N=2

In this section we will construct exact solutions of the family (1) at N = 2. In this case Eq. ( 3) takes the form

Making the substitution (2) into ( 16) and integrating the result we obtain

Substitution of solution ( 4) into (17) allows us to get the following equation

Solving this equation we obtain

Formula ( 4) in the case N = 2 is the following

5 Periodic wave solutions of Eq.( 3) in the case N=3

Formula (1) in the case N = 3 is

Using traveling wave reduction (2), the following ordinary differential equation takes the form

In the case N = 3 from Eq.( 9) we obtain values of the parameters A 3 , B 3

Relations between the coefficients α 0 , α 1 , α 2 and α 3 are the following

Solutions of Eq.( 24) take the form

where values of parameters A 3 , B 3 are given by formulae ( 25) and ( 26).

In the case m = 3, C 0 = 1, z 0 = 0, α 0 = 1 and α 1 = 1 solution of Eq. ( 29) is presented in Fig. 1.

Note that in the case N = C(2l + 1), where C is an arbitrary constant and l = 1, 2, .., solutions (4) are alike solution given in Fig. 1.

Let us look for exact solutions of the family (1) in the case N=4

This equation possesses the traveling wave reduction (2) with y(z) satisfying the equation

Following the procedure suggested in section 2 we obtain values of the parameters A 4 , B 4

and correlations on the coefficients Eq.( 4) takes the form

where A 4 and B 4 are determined by formulae (32) and (33).

In the case m = 2 solution (37) is given in Fig. 2. Note, that the amplitude and period of the traveling wave solution are growing with the increasing of N. The dependence of amplitude on N is illustrated in Fig. 3 in the case m = 2.

for m = -2j+1 2(N -j)-1 , -j N -j (j = 0, 1, …, N -1). P N -1 N are polynomials of N -1 power. Relations for values of the coefficients α k (k = 0, 1, .., N) are found from (9).

Let us formulate shortly the results of this paper. We have studied the generalized K(m, m) equations. Taking into consideration the traveling wave ansatz we have found the periodic wave solutions for a family of n

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