Towards the theory of Benney equations

where

where

From the system (2) we get the relations

and ∂ ∂v g(x, v, t) + ∂ ∂v h(x, v, t) = 0, which is equivalent the nonlinear p.d.e.

is followed.

To construction of particular solutions of this equation can be used the method of the (u,v)transformation developed first by author .

To integrate the partial nonlinear differential equation

can be applied a following method. We use the change of the functions and variables according to the rule

In result instead of the equation (4) one get the relation between the new variables u(x, t, z) and v(x, t, z) and their partial derivatives

In some cases the integration of the last equation is more simple problem than integration of the equation (4).

To illustrate this method let us consider some of examples.

The equation

is transformed into the following form

Using the substitution

we find the equation for ω(x, t) ∂ ∂x ω(x, t) + t 2 = 0.

Its integration lead to

where F 1 (t) is arbitrary function. Now with help of ω(x, t) we find the functions u(x, t) and v(x, t)

After the choice of arbitrary function F 1 (t) and elimination of the parameter t from these relations we get the function z(x, y), satisfying the equation (7).

The equation

is transformed into the following form

Using the substitution

we find the equation for ω(x, t)

Its integration give us the function

where F () is arbitrary function. Now with help of the function ω(x, t) we can find the functions u(x, t) and v(x, t). Then after the choice of arbitrary function F () and elimination of the parameter t from the relations

we can get the function z(x, y), satisfying the equation (8).

By analogy the substitution

into the equation for ω(x, t) lead to the equation

which also give us the solution of the equation (8).

The equation meeting in theory of the Benney equations

after the (u,v)-transformation with the change (y → t) takes the form

In result of substitution of the form

we get from this relation the linear p.d.e. with the respect of the function ω(x, t, z)

Its solutions can be obtained by the Laplace method and after elimination of the parameter t from the expressions for the function (f (x, y, z) → u(x, t, z)) and (y → v(x, t, z)) the solutions of initial equation ( 9) can be constructed.

By analogy this method can be applied to obtaining particular solutions of the equation (3).

3 The (u, v)-transformation of the equation ( 3)

For convenience we write the equation ( 3) in the form

where instead of the variables v and t we used the variables y and z.

After the (u, v)-transformation with the change of the variable y on parameter t the equation (10) takes the form of the relation between the functions u(x, t, z) and v(x, t, z) and their derivatives

There are a lot possibilities to bring this relation to one equation. Classification all types of reductions is open problem.

We use a simplest type of reductions.

As example after the substitution

this relation lead to the nonlinear partial differential equation

Its particular solutions can be used for construction of solutions of the equation (10). Let as consider some examples. Using the substitution ω(x, t, z) = A(t, z) + Bxt (13)

we get from the (12) the equation

with general solution

dependent from two arbitrary functions F2 (z) and F1 ( zB-ln(t)

). In particular case

B 2 with the help of the formulaes (11, 13) we find the relations

From last equation we get the expression for parameter t t = e -LambertW (1/2 B 2 (-y+ F2 (z)+Bx)e zB )+zB and after substitution its into the first one we find the function f (x, y, z)

which is solution of the equation (10

It has the particular solution defined by the expression ω(x, t, z) = P (t, z) + xe -z (16)

where the function P (t, z) satisfies the equation

having general solution dependent from two arbitrary functions

In particular case

we find the solution of the equation ( 10)

Remark 1 The equation ( 15) after the substitution

and can be solved exactly.

In fact, the function B(t, η) = A(t, η) + 1 satisfies the equation

which admits the first integral

where K(η) is arbitrary. Equation ( 18) is in form

After the (u, v) -transformation it is reduced to the relation

which is equivalent the first order nonlinear p.d.e. with respect to the function ω(x, t) at the substitution

Solutions of the equation ( 20) depend from the function K(ω t ).

As example in the case K(ω t ) = ∂ω(x, t) ∂t we get the linear equation

Remark 2 Legendre-transformation of the equation ( 19)

lead to the equation

which is reduced to the Bernoulli equation after application of the suitable Legendre transformation and so it is integrable.

With the help of solutions of the equation (19) the functions A(t, η) and ω(x, t, z) = A(t, xz) = A(t, η) can be determined.

Then after elimination of the parameter t from the expressions for the functions ω(x, t, z) and y(x, t, z) particular solutions of the e

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