Elizabeth smiled, and waved her hand to the solitary figure.
"The Tempting of Tavernake" by E. Phillips Oppenheim
At four o'clock I observed a wild duck swimming on the waves, a single solitary wild duck.
"Biographia Literaria" by Samuel Taylor Coleridge
The seven Englishmen tossing on the waves in this solitary part of the globe would not fail to remember that.
"The Life of Captain Matthew Flinders" by Ernest Scott
The Rector, recognizing the solitary rider, waved his hand in welcome, and Sir Henry pulled up.
"Elizabeth's Campaign" by Mrs. Humphrey Ward
The abandoned harvests waved solitary in the wind, and the numerous water-wheels were all motionless.
"A Narrative of the Expedition to Dongola and Sennaar" by George Bethune English
I am a solitary wave in the dark and desolate sea: and the sparkling glass I drank was drugged with misery.
"Peter Schlemihl" by Adelbert von Chamisso
As a rule I should have waved a "good morning" and passed on for my solitary walk.
"The King's Mirror" by Anthony Hope
Again Frank's jubilant shout echoed oddly clear and solitary above the incessant booming of the breakers and the monotonous wash of the waves.
"Golden Days for Boys and Girls" by Various
In spirit, he is in that solitary house, surrounded by the waves.
"The Green Book" by Mór Jókai
They had no long rambles now in solitary lanes and under flowering hedgerows beside the waving green corn.
"John Marchmont's Legacy, Volumes I-III" by Mary E. Braddon
***
I strode to where I saw him stand
So solitary in the sun -
Nothing! just empty sew and land,
no smallest sign of anyone.
While down below I heard the roar
Of waves, five hundred feet or more.
"My Suicide" by Robert W Service
Next day was told what deeds of night
Were done; the web had vanished quite;
With it the strange opposing pair;
And listless waved on vacant air,
For her adieu to heart's content,
A solitary filament.
"A Garden Idyl" by George Meredith
This last step involves, among other things, the fact that these solitary waves are well approximated by the KdV soliton and the fact that the linearization of the KdV equation about its solitary wave is well understood.
A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam model near the KdV Limit
A transformation of the original FPU equation (2) into a form in which we can prove that its solitary wave solution is close to that of the Toda lattice.
A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam model near the KdV Limit
V (x) = (e−x + x − 1) . (Throughout this paper, quantities with tildes will refer to the Toda model.) Note that ˜V satisﬁes the hypotheses of so the results of that paper imply that the Toda model has a family of solitary waves close to those of the KdV equation.
A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam model near the KdV Limit
Let uc by the proﬁle of the solitary wave of the FPU model (1) with speed c, and let ˜uc be the proﬁle for the special case of the Toda potential.
A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam model near the KdV Limit
The estimates on ξ1 − ˜ξ1 estimate because of the prior remark about the scaling of the ℓ2 then follow from these two since Theorem 1.1 shows that a derivative of the solitary wave proﬁle with respect to the spatial variable gains exactly one power of ǫ.
A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam model near the KdV Limit
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