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New exact solutions to the KdV2 equation (also known as the extended KdV equation) are constructed. The KdV2 equation is a second-order approximation of the set of Boussinesq’s equations for shallow water waves which in first-order approximation yields KdV. The exact solutions

Water waves have attracted the interest of scientists for at least two centuries. One hundred and seventy years ago Stokes [

For the shallow water problem leading to KdV, two small parameters are assumed: wave amplitude/depth

One of reasons for the enormous success of the KdV equation is its simplicity and integrability. However, KdV is derived under the assumption that both

The next, second-order approximation to Euler’s equations for long waves over a shallow riverbed is

Despite its nonintegrability, KdV2 possesses exact analytic solutions. The first class of such solutions, single solitonic solutions found by us, is published in [

In [

This article complements [

Our idea to look for exact solutions to KdV2 in the same forms as solutions to KdV gained recently a strong support by results of Abraham-Shrauner [

In order to present the approach we show the KdV case first. Assume solutions to KdV in the following form:

Introduce

Equations (

Coefficients

Now, we look for solutions to KdV2 (

Assume solutions to KdV2 in form (

Insertion of (

As shown previously in [

Express (

Solving (

Denote

Then volume conservation requires

From properties of elliptic functions

In order to obtain explicit expressions for coefficients

With this choice

Substitution of

Velocity formula (

In general, as stated in previous papers [

Function

Velocity

It is clear that for real-valued

Amplitude

Coefficient

Coefficient

Velocity

For

Velocity formula (

In this case

Amplitude

Coefficient

Coefficient

Velocity

Table

Examples of values of

| | | | | | | |
---|---|---|---|---|---|---|---|

0.10 | 0.10 | 0.99 | 4.108 | 1.5683 | −0.5646 | 1.107 | 9.426 |

0.30 | 0.30 | 0.99 | 1.369 | 0.9054 | −0.1882 | 1.107 | 16.33 |

0.50 | 0.50 | 0.99 | 0.822 | 0.7013 | −0.1129 | 1.107 | 21.08 |

0.30 | 0.30 | 0.80 | 3.028 | 1.3465 | −0.7904 | 1.071 | 6.706 |

0.50 | 0.50 | 0.80 | 1.817 | 1.0430 | −0.4743 | 1.071 | 8.657 |

Figure

Profiles of the KdV2 (blue lines) and KdV solutions (red lines) for

Table

Examples of values of

| | | | | | | |
---|---|---|---|---|---|---|---|

0.30 | 0.30 | 0.10 | −134.5 | 2.105 | 63.83 | 3.170 | 3.064 |

0.50 | 0.50 | 0.05 | −71.44 | 1.535 | 34.82 | 0.717 | 4.147 |

In Figure

Profiles of the KdV2 solutions for

In the cases

For

Numerical calculations of the time evolution of superposition solutions performed with the finite difference code as used in previous papers [

Time evolution of the superposition solution

The case corresponding to

Time evolution of the superposition solution

From periodicity of the Jacobi elliptic functions it follows that

From the studies on the KdV2 equation presented in this paper and in [

There exist several classes of exact solutions to KdV2 which have the same form as the corresponding solutions to KdV but with slightly different coefficients. These are solitary waves of the form

KdV2 imposes one more condition on coefficients of the exact solutions than KdV.

Periodic solutions for KdV2 can appear in two forms. The first form,

All the above-mentioned solutions to KdV2 have the same function form as the corresponding KdV solutions but with slightly different coefficients.

KdV, besides having single solitonic and periodic solutions, possesses also multisoliton solutions. The question of whether exact multisoliton solutions for KdV2 exist is still open. However, numerical simulations presented in the Appendix, in line with the Zabusky-Kruskal numerical experiment [

For KdV there exist multisoliton solutions which can be obtained, for example, using the inverse scattering method [

Emergence of soliton trains according to KdV from initial cosine wave.

Emergence of soliton trains according to KdV2 from initial cosine wave.

Initial conditions for both simulations were chosen as hump

In a multisoliton solution of KdV each soliton has a different amplitude. Otherwise these amplitudes are arbitrary. If multisoliton solutions to KdV2 exist we would expect some restrictions on these amplitudes.

The authors declare that they have no conflicts of interest.