More Plasma Physics: Looking At Solitons And Soliton Solutions (Of The Korteweg deVries Equation)

 

In space plasma physics many situations arise in which one will wish to examine the behavior of a solitary wave - say in a particular plasma environment. One such is for nonlinear ion-acoustic waves. 

 The solitary wave (soliton) is just a traveling wave of form:  

u(x, t) =  f(x - ct)  

for some smooth function f that decays rapidly at infinity and under certain conditions can travel without change of shape (see graphic). 

  Consider the partial differential equation:

 u_tt   - u_xx  =   u ( 2 u²  -  1)

Which has a family of solitary wave solutions:

u(x,t) = sech (x cos q +   t sinh q)

We do not expect, for mathematical reasons,  the sum of two such solutions (superposition) to also be a solution.  There is, however, a class of soliton equations which does exhibit a form of nonlinear superposition.  An n-soliton solution then is a solution that is asymptotic to a nontrivial sum of n solitary waves, e.g.

å ⁿ_t=1 f _i  (x - c_i  t) as t -> - oo  and to the sum of the same waves:

å ⁿ_t=1 f _i  (x - c_i  t  + r_i)

with non-zero phase shifts r_i  as t -> oo.  I.e. after nonlinear interaction the individual solitary waves pass through each other, keeping their velocities and shapes - but with phase shifts. 

 The  Korteweg de Vries (KdV) equation:

(- v _o  + c _s   + v) ¶  v / ¶x’  -  m ¶ ² v / ¶x²’  

+  a ¶ ³ v / ¶x³ =  0

is a well known example of  such a soliton equation admitting nonlinear superposition.  Here c _s is the ion sound speed, and v _o the  electron thermal speed.  In the form shown we note the appearance of the dissipative term (m ¶ ² v / ¶x²)  and that for a soliton to evolve into a shock a dissipative mechanism is needed.  In the more common situation steepening of the wave balances dispersion and one will obtain a wave  form such a shown in the top graphic. We focus more on this case now by adjusting the KdV equation.

  On integrating once,  and excluding the shock evolution term (in m) ,  we obtain:

a  d² V/ dx²   -    (v _o   -   c _s ) v  -   v ²/ 2   =  0 

Which has the same mathematical form as Newton's 2nd law of motion, e.g.  m x" =  F(x)  =   -  ¶ x V(x)  

Where V(x) is the potential energy. With some further manipulation we find:

a d² V/ dx²   =   -  ¶ v  V' (x)  =

-  ¶ v [(c _s   -  v _o) v ²/ 2   +     v³/ 6 ]

For a particle 0f mass a   moving under a potential field given by the quantity in brackets.  We call this quantity the pseudo potential and designate it:

 F (v) =  [(c _s   -  v _o) v ²/ 2   +     v³/ 6 ]

Which is also known as the Sagdeev potential.  It is left to the industrious reader to do a simple plot of  F (v) vs.  v,   with   v _max   shown on v -axis.

For the criteria on  F  to obtain a soliton solution we have:

i)   ¶ F / ¶ V ] _{v= 0}  =  0;  ¶ ² F/ ¶ v²   <   0

ii)  F   <   0,   For   0   <   v   <   v _max

iii)  d F / d V ] _{v= v max}  >  0

We  note here that two graphs of  F   vs.  v are possible,  one for  c _s   -  v _o   >   0,  the other for  c _s   -  v _o   <   0.  Since we demand only a localized wave form then it will always be the latter form used, i.e. in further analyses.  One such is to obtain a soliton solution for the KdV equation:

a  d² V/ dx²   -    (v _o   -   c _s ) v  -   v ²/ 2   =  0 

This may be solved exactly i.e. with c _s   -  v _o   <   0.   The procedure is then to multiply the KdV by v'  and then integrate to obtain:

a/ 2  (v' ²)  =  (v _o   -   c _s ) v ²/ 2  -    v³/ 6

And we choose the constant of integration to be zero because we want v' = 0  when  v = 0.   Working through the process the final solution is found to be:

v   =   3 (v _o   -   c _s ) sech ²  [(v _o   -   c _s  / 4 a) ½     x' ]

Suggested Problems:

1)  Plot a graph of  F   vs.  v  for the case c _s   -  v _o   <   0 and indicate the position of v _max  on the graph.

2)  Integrate  the KdV equation:

a  d² V/ dx²   -    (v _o   -   c _s ) v  -   v ²/ 2   =  0 

And show how the soliton solution:

v   =   3 (v _o   -   c _s ) sech ²  [(v _o   -   c _s  / 4 a) ½     x' ]

Is obtained.  Given this is in the fluid frame, show what it would be in the lab frame.

FLA Gubernator DeSantis Resorts To Strong Arm State Law To Dictate University Education

 "This is the last gasp of a once powerful party that is intellectually impotent.   And when you have an intellectually impotent ideology or party it will cheat.  When you can't compete in the marketplace of ideas you fundamentally do what DeSantis is doing.  It is market protectionism.  It doesn't want to compete about learning about poverty, or the history of this nation, or global capitalism so it wants to cheat.  So it must endorse what is inconsistent with intellectualism."-   Prof.  Tressie McMillan Cotton, ALL In, June 23.

"There is a real thing Governor Ron DeSantis is putting his finger on:  People go into these colleges and come out liberals.  What's goin' on? Except his solution for it is a perfect example of the kind of cultish authoritarianism in the modern Republican party under Donald Trump. They have a political problem - how to attract college-educated voters  But the solution they come up with is to just take the sledge hammer of the state to the institution of higher education." - Chris Hayes, on ALL In,  June 23

"All liberal arts education is defined in consciously non-vocational terms.  It is not a preparation for this job or that, for one career rather than another.  It is a preparation for the job of living, which of course is not a job at all.   The variety of liberal arts education is enormous but all rest on the assumption that one important aim of undergraduate education is to afford the young who are its beneficiaries the opportunity to reflect on the curious and inspiring adventure of life before they have gone too far into it."  - Anthony Kronman,  'Education's End - Why Our Colleges and Universities Have Given Up On The Meaning Of Life."  (p. 41)

Florida Gov. Ron DeSantis signed several education-focused bills last week, including one to "protect"  what the state calls the “expression of diverse viewpoints” at state colleges and universities.  This will be by surveying students on how they feel their "freedoms"  are protected on campus.

Under HB 233, signed by DeSantis on June 22, the Florida Board of Education will now require Florida colleges to survey students every year and assess “the intellectual freedom and viewpoint diversity” at each institution.  Translation:  How much conservo crap are the colleges willing to allow to enable more young brains to become young Trumpers?

The FLA Board of Education  will also be required to select or create an "objective, nonpartisan and statistically valid survey"  for each institution to use, and determine the extent that competing ideas are presented to the college’s community. The survey will check how free students (i.e. young Reeps)  feel to express their own beliefs and viewpoints while on campus and in the classroom.

In addition, the new law adds the new term “shield” which, in this context, the state has defined as “to limit students,’ faculty members’ or staff members’ access to, or observation of, ideas and opinions that they may find uncomfortable, unwelcome, disagreeable, or offensive.”  

All sounds pretty tame and justifiable, right?  Wrong!  It's an authoritarian Reep Guv Nazi wannabe-  who fancies being the next Trump -  taking a sledge hammer to higher education in the state - as per Chris Hayes' quote at the top.  How?  By using the threat of budget cuts to those universities, colleges that don't  come out with the desired results in his mandated "surveys".     

Why?  Because he sees that too many leaving university now are the most left-leaning in a generation. (Only 26% leaving university in a recent AEI survey describe themselves as "conservative').  Instead of recognizing this as part and parcel of true critical thinking and being exposed to a wide spectrum of ideas, he sees it as indoctrination.   He also sees, is aware, that our politics is increasingly divided along the lines of education - and he doesn't like it.

Even when you control for other factors, like race, creed, income etc.people with higher levels of education are found to be more liberal.   Those with less education are more conservative,  This is why those who lean Republican say in polls that college is having a negative effect on the country, the percentage going from 37% (in 2015) to 59% in 2019.   This divide is particularly sharp among white people.   So mindful of all this Ronny DeSantis believes taking a sledge hammer to college studies in Florida can at least slow it down.  Of course, he's a fool.

Which explains his proposed survey to "seek out ideas and opinions they may find offensive".   Yeah, like critical race theory or basic Marxism - which might cause fragile young conservative brains to explode.  So these surveys will be a way to keep profs under control from contaminating too many susceptible minds, threatening their colleges with budget cuts if the don't adhere to state sanctioned twaddle in curricula -   specially in political and social science.

As per The Tampa Bay Times, June 22: "Budget cuts could take place if the colleges and universities are found to be indoctrinating their students".   Which, as Hayes put it, "might be the case if those surveys don't produce the right answers for DeSantis and his buddies."   In effect, it amounts to an outrageous abuse of state power by surveilling citizens' thoughts, learning. Threatening to punish them if they disclose the "wrong" views.  It sounds like attending to free speech but in fact is the very opposite of that, akin to book burnings in the Third Reich.   

As Hayes noted, DeSantis mask of B.S. fell off during one news conference,  showing his yen to suppress free speech. There we see DeSantis blabbering:

"I know a lot of parents, one of the things they worry about is you send a kid to university and are they basically gonna be indoctrinated. Are they actually gonna be taught to think for themselves, challenge assumptions and be critical thinkers and learners.  We obviously want out universities to be focused on critical thinking and academic rigor.  We do not want then  not hotbeds for stale ideology."   

Focused on critical thinking?   You semi-educated white ape.  Didn't you know that includes exposure to ideas, beliefs and yes, ideologies, outside of one's comfort zone?  Real critical thinking - not the middle mind vanilla substitute on offer from Ron "Goebbels"  DeSantis  - means being offered fora with radically different themes, ideas.  Like when Loyola's Jesuits invited atheist existentialist Jean Paul-Sartre to speak at Loyola in January, 1966, 

That exposure began my path to atheism- materialism and I can thank Loyola's Jesuits for it!

DeSantis, dope that he is, would probably call that indoctrination.  But  the Jesuits didn't indoctrinate.  They simply allowed an intellectual stage for diverse thought, and encouraged interested students to attend.  Obviously-  as Hayes noted-  DeSantis doesn't think it's indoctrination to teach kids that communism is bad, i.e. as a state mandate.  But in fact that amounts to capitalist indoctrination.

Blogger and professor Henry Giroux has noted:

"Many see higher education as "dangerous" because its facilities hold the potential to serve as laboratories for democracy where students learn to think critically. Teachers are threatening because they refuse to conflate education with training or treat schools as if they were car dealerships. Many educators have made it clear that they regard teaching for the test and defining accountability only in numerical terms as acts that dull the mind and kill the spirit of students. Such repressive requirements undermine the ability of teachers to be creative, engage with the communities in which they work and teach in order to make knowledge critical and transformative. The claim that we have too many bad teachers is too often a ruse to hide bad policies and to unleash assaults on public schools by corporate-driven ideologues and hedge fund managers who view schools strictly as investment opportunities for big profits."

Adding:

"Neoliberal education is increasingly expressed in terms of austerity measures and market-driven ideologies that undermine any notion of the imagination, reduce faculty to an army of indentured labor and burden students with either a mind-numbing education or enormous crippling debt or both. 

If faculty and students do not resist this assault, they will no longer have any control over the conditions of their labor, and the institutions of public and higher education will further degenerate into a crude adjunct of the corporation and financial elite."

Which is exactly what the likes of Ron DeSantis really wants: marginalization of citizens' choices and life quality and control of their thoughts. Especially if there's a chance of creating more Reeps and Trumpies to help downgrade the national status into an even more degraded "adjunct" of the financial elites and white nationalist scum.  Especially, if their biggest 'wet dream' is consummated, and Trump somehow manages to get elected again in '24.

See Also:

Academic Freedom | Constitution, Civil Rights, Civil Liberties

Florida's Shameful, Recent Past Echoes in Desantis War on Academic Freedom

by Will Bunch | July 1, 2021 - 6:50am | permalink

— from The Philadelphia Inquirer

And:

Academic Freedom

A Dangerous Moment for Academic Freedom

by Jill Richardson | July 2, 2021 - 6:20am | permalink

— from OtherWords

And:

Racial Justice

I'm a Scholar of Critical Race Theory — Here's the Reality about It Behind the Conservative Moral Panic

by Mia Brett | June 29, 2021 - 7:44am | permalink

— from Alternet

AND:

Socialism

Does Socialism Have a Public Relations Problem?

by Sonali Kolhatkar | June 29, 2021 - 6:27am | permalink

What The Pentagon Prelim UFO Report Isn't Saying: 'The Aliens Are Here, We Just Don't Want To Admit It."

 

"I think in all likelihood this report will come and go, and with it the mainstream chatter around U.F.O.s, until definitive proof is exposed. A planet that can’t come together on climate change or a global pandemic might not pay much attention even if wreckage or an alien corpse is discovered. The culture wars alone might eclipse it, so rabidly are we in their grips." - Chris Carter, NY Times, Saturday   

U.S. intelligence officials reviewing dozens of reports of mysterious flying objects found 18 in which the objects displayed no visible propulsion or appeared to use technology beyond the known capabilities of the U.S. or its adversaries.  This according to an intelligence report released Friday.  

But this may not have been immediately apparent to  the casual reader wading through nine pages of dodgy techno babble, much of it replete with too many head -spinning qualifiers and cautions - as we will see.  One example of what I mean: According to one official connected with the report:  

"We have no clear indications that there is any nonterrestrial explanation,” 

But as the NY Times' Chris Carter  opined, he might have said exactly the same thing if wreckage of an alien craft had been discovered, or an alien corpse.  If the aim is to "terrestrialize" any findings (i.e.  overthinking them into an earthly or mundane frame), then the refrain will always be that:  "greater technological understanding may be required to determine what was behind the unexplained cases."  

My point?  If the unexplained craft seen in the videos are as advanced as I suspect then even a thousand more years of human techno development may not be sufficient to solve the riddle of their propulsion.  Hence, the appeal to a "greater technological understanding" right now emerges as just a clever copout. It is tech speak 'filler' designed to avoid the necessary conclusion these are alien craft, but "We have no idea what to do about them because we can't match their technology."  Such an admission would be more honest but would freak out the terrestrialist Senators - like Marco Rubio -  who pushed for the release of the report (because he suspects the UAP  are Chinese or Russian craft).   

Given this background let's look at some of the main parts and offer some interpretations, insights:

"Most of the UAP reported probably do represent physical objects given that a majority of UAP were registered across multiple sensors, to include radar, infrared, electro-optical, weapon seekers, and visual observation. In a limited number of incidents, UAP reportedly appeared to exhibit unusual flight characteristics. 

These observations could be the result of sensor errors, spoofing, or observer misperception and require additional rigorous analysis. There are probably multiple types of UAP requiring different explanations based on the range of appearances and behaviors described in the available reporting."

Here I've separated the straight up honest assessment (top paragraph) from the hooey. By that I mean the acknowledgement of the use of multiple sensors rules out sensor errors or spoofing.  This is critical and I will come back to it again and again.  It is simply damned near impossible to simultaneously rig multiple sensors to deliver abnormal findings - all of which show the same intelligent control of the craft. Thus, the appeal to "sensor errors" and "observer misperceptions" in the 2nd paragraph is merely intended as a distraction from the main point.  Much like Neal deGrasse Tyson's "glitch" nonsense.  

We go again to the more honest segment describing the core signal:

 "A Handful of UAP Appear to Demonstrate Advanced Technology In 18 incidents, described in 21 reports, observers reported unusual UAP movement patterns or flight characteristics. Some UAP appeared to remain stationary in winds aloft, move against the wind, maneuver abruptly, or move at considerable speed, without discernable means of propulsion. In a small number of cases, military aircraft systems processed radio frequency (RF) energy associated with UAP sightings. The UAPTF holds a small amount of data that appear to show UAP demonstrating acceleration or a degree of signature management. 

Additional rigorous analysis are necessary by multiple teams or groups of technical experts to determine the nature and validity of these data We are conducting further analysis to determine if breakthrough technologies were demonstrated. 

But as I already noted, if the unexplained craft seen in the videos are as advanced as I suspect then no amount of "more rigorous analysis" now will reveal their nature or underlying breakthrough technologies.   This is because it takes a more advanced technology to adequately analyze a given breakthrough  technology.  Think of Neanderthals trying to make sense of an F -117 Stealth aircraft and you get the idea.   Or in the words of another official behind the report:  "Some of those could require scientific advances on our part to allow us to better understand what it is that we’re observing,”

The point?  No current terrestrial technology (even hypersonic)  exists to do that and will be unlikely to attain such level even in another thousand years as I pointed out.  The inescapable fact is the "breakthrough technologies" are already self evident given we can't duplicate them.  Why not just admit it and also that we are seeing indirect evidence for extraterrestrial craft in our airspace? 

Perhaps the best answer has been provided in a paper published in the journal Political Theory by Alexander Wendt and Raymond Duvall, viz.

Sovereignty and the UFO - Alexander Wendt, Raymond Duvall, 2008

Herein they note the phenomenon of the UFO tends to be rejected as real because it comes up against the human concept of state sovereignty. Prof. Daniel W. Dresner, in an article appearing in The Sunday Denver Post on June 2, 2019, put it this way:

'UFOs have never been systematically investigated by science or the state, because it is assumed to be known that none are extraterrestrial. Yet in fact this is not known, which makes the UFO taboo puzzling given the ET possibility.  The puzzle is explained by the functional imperatives of anthropocentric sovereignty, which cannot decide a UFO  exception to anthropocentrism  while preserving the ability to make such a decision. The UFO can be known only by not asking what it is."

Hold that thought: a "UFO exception to anthropocentrism"  - in other words, an entity regarded seriously as an exception to human superiority. Hence, the real reason UFOs have been dismissed or treated as unworthy of serious human attention is because of the existential challenge they pose to a species that fancies itself the lords of creation - as well as the most technologically advanced life form.  In effect, to admit the reality of the UFO as an alien craft would directly challenge this comforting codswallop, engendering enormous cosmic uncertainty and rejection of the fake techno superiority meme.    This is also exactly why "stigmas"  arise associated with UFO research or claims/admissions of their reality.  In a word, those  skeptical responses amount to an elaborate defense mechanism for egocentric humans, who would otherwise have to fess up that they're second rate at least in technological terms.  We see this even in the released  report, e.g.

"Narratives from aviators in the operational community and analysts from the military and IC describe disparagement associated with observing UAP, reporting it, or attempting to discuss it with colleagues. 

Although the effects of these stigmas have lessened as senior members of the scientific, policy, military, and intelligence communities engage on the topic seriously in public, reputational risk may keep many observers silent, complicating scientific pursuit of the topic. "

We also see this avoidance dynamic in assorted distractive baloney and balderdash in other parts of the report, e.g.

"Our analysis of the data supports the construct that if and when individual UAP incidents are resolved they will fall into one of five potential explanatory categories: airborne clutter, natural atmospheric phenomena, USG or U.S. industry developmental programs, foreign adversary systems, and a catchall “other” bin. UAP clearly pose a safety of flight issue and may pose a challenge to U.S. national security."

The five categories "construct"   is again,  a lot of hooey- mental fluff and piffle designed to distract from serious inquiry.  There are not any "five  categories" only one real one that's directly associated with those UAP  "beyond U.S. or its adversaries technological capabilities" described at the outset.  These form the very core of the UAP which point to an alien origin, without the need to drag in gigabytes of bunkum and artificially massaged mundane window dressing.  

While "experts" supposedly continue to check sensors for "flaws" to explain the extraordinary dynamical behavior and one skeptic clown, Mike West(who actually admitted - D. Post, 6/23-   that alien craft was the simplest hypothesis) has sought to invoke "infrared glare" to account for certain rapid spin motions - the simplest explanation is avoided at all costs.  Why? Because humans would prefer to overthink, invent outlandish pseudo "mundane" explanations and make excuses rather than admit we are not at the top of the cosmic food chain we had supposed. In the words of Prof. Dresner:   

"The UFO can be known only by not asking what it is."

In other words, dodge the reality by covering it in bunkum,  techno babble and bland, middle mind tropes and memes. Or to quote astronomer Chris Impey:  "This is still mostly an area where scientists fear to tread."   

Well, you can add high end public officials, most of the corporate media and the military too.  In the meantime it's best to remember the words of one of the most beloved literary sleuths:  "When you've eliminated everything else whatever remains - however unlikely - must be the answer."

See Also:

Even if You Think Discussing Aliens Is Ridiculous, Just Hear Me Out     

How would contact with U.F.O.s and other civilizations change ours?

And:

Preliminary Assessment: Unidentified Aerial Phenomena