How Long to Read Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves

How Long Does it Take to Read Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves?

It takes the average reader 2 hours and 26 minutes to read Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves by GŽrard Iooss

Assuming a reading speed of 250 words per minute. Learn more

Description

The authors consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta$ between them. Denoting by $\mu =gL/c^{2}$ the dimensionless bifurcation parameter ( $L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), bifurcation occurs for $\mu = \cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. ``Diamond waves'' are a particular case of such waves, when they are symmetric with respect to the direction of propagation. The main object of the paper is the proof of existence of such symmetric waves having the above mentioned asymptotic expansion. Due to the occurence of small divisors, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta$, the 3-dimensional travelling waves bifurcate for a set of ``good'' values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane $(\theta,\mu ).$

How long is Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves?

Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves by GŽrard Iooss is 144 pages long, and a total of 36,576 words.

This makes it 49% the length of the average book. It also has 45% more words than the average book.

How Long Does it Take to Read Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves Aloud?

The average oral reading speed is 183 words per minute. This means it takes 3 hours and 19 minutes to read Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves aloud.

What Reading Level is Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves?

Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves is suitable for students ages 10 and up.

Note that there may be other factors that effect this rating besides length that are not factored in on this page. This may include things like complex language or sensitive topics not suitable for students of certain ages.

When deciding what to show young students always use your best judgement and consult a professional.

Where Can I Buy Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves?

Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves by GŽrard Iooss is sold by several retailers and bookshops. However, Read Time works with Amazon to provide an easier way to purchase books.

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