Benjamin-Feir instability

In his 1847 paper, G.G. Stokes proposed the existence of periodic wave-trains in non-linear systems. In the case of waves on deep water, the first two terms in the asymptotic expansion employed by Stokes are given by

$$ \eta ( x,t ) = a \cos ( \zeta ) + { \frac{1}{2} } ka ^ {2} \cos ( 2 \zeta ) , $$

where $ \zeta = kx - \omega t $ and $ \omega ^ {2} = gk ( 1 + k ^ {2} a ^ {2} ) $. Not everyone was convinced that the series converges and therefore that periodic waves actually exist. Convergence of the series for waves on infinitely deep water was finally established in 1925 by T. Levi-Civita and extended the next year to waves on water of finite depth by D.J. Struik (a brief history is given by [a4]). With the existence of the Stokes wave established, it is quite remarkable that no one noticed its instability, until T.B. Benjamin and J.E. Feir during the 1960s (see, e.g., [a4], [a5]).

Adding perturbations with frequencies close to the carrier frequency $ \omega $, of the form

$$ \epsilon ( x,t ) = \epsilon _ {+} { \mathop{\rm exp} } ( \Omega t ) \cos [ kx - \omega ( 1 + \delta ) t ] + $$

$$ + \epsilon _ {-} { \mathop{\rm exp} } ( \Omega t ) \cos [ kx - \omega ( 1 - \delta ) t ] , $$

Benjamin and Feir used a linearized analysis to show that

$$ \Omega = { \frac{1}{2} } \delta ( \sqrt {2k ^ {2} a ^ {2} - \delta ^ {2} } ) \omega. $$

Thus, the perturbations grow exponentially provided

$$ 0 < \delta < \sqrt 2 ka . $$

It is important to note that the instability is controlled by the wave-number $ k $ and the amplitude $ a $ of the carrier wave (larger values of $ ka $ allow more unstable "modes" $ \delta $, thus enhancing the instability).

One practical implication of the Benjamin–Feir instability is the disintegration of periodic wave-trains on sufficiently deep water, as demonstrated by their wave tank experiments (see, e.g., [a4]). However, it was experimentally observed [a10] that the periodic wave need not always disintegrate (under certain circumstances the instability may lead to a Fermi–Pasta–Ulam-type recurrence). This means that the Benjamin–Feir instability saturates and that the initial wave-form is (approximately) regained after a while.

This whole process is best understood by investigating the normalized non-linear Schrödinger equation,

$$ iA _ {t} + A _ {xx } + 2 \left | A \right | ^ {2} A = 0, $$

which describes the evolution of weakly non-linear wave envelopes on deep water, among other things (for a general introduction see [a2]). This equation also has the distinction that it is completely integrable and that it allows soliton solutions. H.C. Yuen and B.M. Lake [a10] observed soliton interactions in wave tank experiments, which demonstrates that the non-linear Schrödinger equation provides a qualitatively satisfactorily description of the long-time evolution of wave packets.

The non-linear Schrödinger equation also exhibits the Benjamin–Feir instability and was used in [a9] to study the long-time evolution of the instability. Numerical experiments indicate that the instability leads to two different types of recurrence, simple and complex, depending on the number of unstable modes.

Subsequently, the Benjamin–Feir instability and recurrence have been associated with homoclinic manifolds (see, e.g., [a6], [a3]) with the complexity of the manifolds increasing with the number of unstable modes. Careful numerical experiments with the non-linear Schrödinger equation show that recurrence is possible for a low number of unstable modes. However, if the number of unstable modes is increased, the non-linear Schrödinger equation becomes effectively chaotic (see [a1]). In practical terms this means that the periodic wave may disintegrate into chaos.

Although soliton interactions and recurrence have been observed in wave tank experiments, as described by the non-linear Schrödinger equation, it remains a challenge to observe homoclinic manifolds experimentally.

References

[a1]	M.J. Ablowitz, C.M. Schober, "Effective chaos in the nonlinear Schrödinger equation" Contemp. Math. , 172 (1994) pp. 253–268
[a2]	M.J. Ablowitz, H. Segur, "Solitons and the inverse scattering transform" , SIAM (1981)
[a3]	M.J. Ablowitz, B.M. Herbst, "On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation" SIAM J. Appl. Math. , 50 (1990) pp. 339–251
[a4]	T.B. Benjamin, "Instability of periodic wavetrains in nonlinear dispersive systems" Proc. Roy. Soc. A , 299 (1967) pp. 59–75
[a5]	T.B. Benjamin, J.E. Feir, "The disintegration of wave trains on deep water. Part 1. Theory" J. Fluid Mech. , 27 (1967) pp. 417–430
[a6]	N. Ercolani, M.G. Forest, D.W. McLaughlin, "Geometry of the modulational instability Part III: homoclinic orbits for the periodic sine-Gordon equation" Physica D , 43 (1990) pp. 349–384
[a7]	B.M. Lake, H.C. Yuen, H. Rungaldier, W.E. Ferguson, "Nonlinear deep-water waves: theory and experiment Part 1. Evolution of a continuous wave train" J. Fluid Mech. , 83 (1977) pp. 49–74
[a8]	G.G. Stokes, "On the theory of oscillatory waves" Cambridge Trans. , 8 (1847) pp. 441–473
[a9]	H.C. Yuen, W.E. Ferguson, "Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation" Phys. Fluids , 21 (1978) pp. 1275–1278
[a10]	H.C. Yuen, B.M. Lake, "Nonlinear deep water waves: Theory and experiment" Phys. Fluids , 18 (1975) pp. 956–960