File:BBM equation - overtaking solitary waves animation.gif
Summary
An animation of the overtaking of two <a href="https://en.wikipedia.org/wiki/solitary_wave_(water_waves)" class="extiw" title="en:solitary wave (water waves)">solitary waves</a> according to the <a href="https://en.wikipedia.org/wiki/Benjamin%E2%80%93Bona%E2%80%93Mahony_equation" class="extiw" title="en:Benjamin–Bona–Mahony equation">Benjamin–Bona–Mahony (BBM) equation</a>. The <a href="https://en.wikipedia.org/wiki/wave_height" class="extiw" title="en:wave height">wave heights</a> of the solitary waves are 1.2 and 0.6, respectively, and their <a href="https://en.wikipedia.org/wiki/phase_speed" class="extiw" title="en:phase speed">celerities</a> are 1.4 and 1.2.
The upper graph is for a <a href="https://en.wikipedia.org/wiki/frame_of_reference" class="extiw" title="en:frame of reference">frame of reference</a> moving with the average celerity of the solitary waves. The <a href="https://en.wikipedia.org/wiki/envelope_(waves)" class="extiw" title="en:envelope (waves)">envelope</a> of the overtaking waves is shown in grey: note that the maximum wave height reduces during the interaction.
The lower graph shows the <a href="https://en.wikipedia.org/wiki/oscillation" class="extiw" title="en:oscillation">oscillatory</a> tail produced by the interaction, showing that the solitary wave solutions of the BBM equation are not <a href="https://en.wikipedia.org/wiki/soliton" class="extiw" title="en:soliton">solitons</a>.
The presented solutions https://wikimedia.org/api/rest_v1/media/math/render/svg/84e6a2bc1b46671c8ec2bd22f48f53e4aa6b9597" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:6.405ex; height:2.843ex;" alt="u(x,t)"> are for the non-dimensional BBM equation of the form: <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c75965667d3c9194d63b953abe2af10d4acc80fe" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:26.114ex; height:2.509ex;" alt="u_{t}+u_{x}+uu_{x}-u_Template:Xxt=0."> The associated solitary waves are of the form: <img src="
https://wikimedia.org/api/rest_v1/media/math/render/svg/e46718f91954f6cba7cf0b97fb5276209145cbf0" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -2.505ex; width:46.781ex; height:6.343ex;" alt="u=3(\nu -1)\operatorname {sech}^{2}\left({\frac 12}{\sqrt {{\frac {\nu -1}{\nu }}}}\left(x-\nu t-x_{0}\right)\right),"> <img src="
with https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.338ex; width:1.243ex; height:1.676ex;" alt="\nu "> the velocity of the solitary wave, <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26408f0658b0a7d599fa6c226a773553784b16b5" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:9.456ex; height:2.843ex;" alt="\nu \notin [0,1],"> sech the <a href=" <img src="https://en.wikipedia.org/wiki/hyperbolic_secant" class="extiw" title="en:hyperbolic secant">hyperbolic secant function</a> and https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:2.402ex; height:2.009ex;" alt="x_{0}"> a horizontal shift of the solitary wave. <img src="
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 20:12, 8 January 2017 | 416 × 160 (600 KB) | 127.0.0.1 (talk) | An animation of the overtaking of two <a href="https://en.wikipedia.org/wiki/solitary_wave_(water_waves)" class="extiw" title="en:solitary wave (water waves)">solitary waves</a> according to the <a href="https://en.wikipedia.org/wiki/Benjamin%E2%80%93Bona%E2%80%93Mahony_equation" class="extiw" title="en:Benjamin–Bona–Mahony equation">Benjamin–Bona–Mahony (BBM) equation</a>. The <a href="https://en.wikipedia.org/wiki/wave_height" class="extiw" title="en:wave height">wave heights</a> of the solitary waves are 1.2 and 0.6, respectively, and their <a href="https://en.wikipedia.org/wiki/phase_speed" class="extiw" title="en:phase speed">celerities</a> are 1.4 and 1.2. <br>The upper graph is for a <a href="https://en.wikipedia.org/wiki/frame_of_reference" class="extiw" title="en:frame of reference">frame of reference</a> moving with the average celerity of the solitary waves. The <a href="https://en.wikipedia.org/wiki/envelope_(waves)" class="extiw" title="en:envelope (waves)">envelope</a> of the overtaking waves is shown in grey: note that the maximum wave height reduces during the interaction.<br>The lower graph shows the <a href="https://en.wikipedia.org/wiki/oscillation" class="extiw" title="en:oscillation">oscillatory</a> tail produced by the interaction, showing that the solitary wave solutions of the BBM equation are not <a href="https://en.wikipedia.org/wiki/soliton" class="extiw" title="en:soliton">solitons</a>. <br>The presented solutions <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>u</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle u(x,t)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e6a2bc1b46671c8ec2bd22f48f53e4aa6b9597" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:6.405ex; height:2.843ex;" alt="u(x,t)"></span> are for the non-dimensional BBM equation of the form: <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow></msub><mo>−<!-- − --></mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mi>x</mi><mi>t</mi></mrow></msub><mo>=</mo><mn>0.</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0.}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c75965667d3c9194d63b953abe2af10d4acc80fe" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:26.114ex; height:2.509ex;" alt="u_{t}+u_{x}+uu_{x}-u_{{xxt}}=0."></span> The associated solitary waves are of the form: <br> <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>u</mi><mo>=</mo><mn>3</mn><mo stretchy="false">(</mo><mi>ν<!-- ν --></mi><mo>−<!-- − --></mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>sech</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo><!-- --></mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mrow class="MJX-TeXAtom-ORD"><msqrt><mfrac><mrow><mi>ν<!-- ν --></mi><mo>−<!-- − --></mo><mn>1</mn></mrow><mi>ν<!-- ν --></mi></mfrac></msqrt></mrow><mrow><mo>(</mo><mi>x</mi><mo>−<!-- − --></mo><mi>ν<!-- ν --></mi><mi>t</mi><mo>−<!-- − --></mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle u=3(\nu -1)\operatorname {sech} ^{2}\left({\frac {1}{2}}{\sqrt {\frac {\nu -1}{\nu }}}\left(x-\nu t-x_{0}\right)\right),}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46718f91954f6cba7cf0b97fb5276209145cbf0" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -2.505ex; width:46.781ex; height:6.343ex;" alt="u=3(\nu -1)\operatorname {sech}^{2}\left({\frac 12}{\sqrt {{\frac {\nu -1}{\nu }}}}\left(x-\nu t-x_{0}\right)\right),"></span> <br>with <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>ν<!-- ν --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \nu }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.338ex; width:1.243ex; height:1.676ex;" alt="\nu "></span> the velocity of the solitary wave, <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>ν<!-- ν --></mi><mo>∉<!-- ∉ --></mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \nu \notin [0,1],}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26408f0658b0a7d599fa6c226a773553784b16b5" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:9.456ex; height:2.843ex;" alt="\nu \notin [0,1],"></span> sech the <a href="https://en.wikipedia.org/wiki/hyperbolic_secant" class="extiw" title="en:hyperbolic secant">hyperbolic secant function</a> and <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.671ex; width:2.402ex; height:2.009ex;" alt="x_{0}"></span> a horizontal shift of the solitary wave. |
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