Looking Again At The Korteweg de Vries Equation

 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 a soliton equation admitting nonlinear superposition, see e.g. a graphical representation of a soliton solution here:

In the KdV equation c _s is the ion sound speed, and v _o the  electron thermal speed.  In the form shown, note the appearance of the dissipative term:

 (m ¶ ² v / ¶x²)  

and 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 we obtain a wave form such as shown in the graphic

  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. 

Forget Cordyceps - The Fungus Featured In 'Last Of Us' - We Have A REAL Killer Fungus To Deal With

 

                               Fungus-eaten head of survivor from cordyceps
                                            

The HBO apocalypse series 'The Last of Us' is definitely one of kind in terms of medically-inspired horror shows, based as it is on a supposedly ferocious fungus (Ophiocordyceps unilateralis, or cordyceps) capable of literally eating away human flesh.  Just the scenes of a giant fungus-ravaged head  (see above) are enough to make the viewer squirm, even one accustomed to 'The Walking Dead'.  Never mind the cordyceps' victims aren't actually zombies - just fungus eaten humans. Further, this biological entity cordyceps featured in 'The Last of Us'  is based on a real, actually existing fungus. 

But, and this is the key point, it is not capable of infecting humans- only assorted ant and spider species. Furthermore, its use is actually common in some health supplements and in Chinese herbal medicines   This is given it can improve immunity and possibly be helpful in fighting cancer cells or shrinking tumor size. Other potential health benefits include athletic performance improvements with stamina and strength, and better kidney function. 

Humans cannot carry the fungus cordyceps or be infected by it in any way due to the high internal temperatures of the human body.  Bottom line, even if the thing did somehow get into a human, the natural body temperature of 98.6F is more than it could endure. And it would have to undergo one helluva evolutionary trick to survive and wipe us out like it does in the series.

 

This elicits the question of why the producers and show runners didn't enlist a genuine fungus threat: Candida auris.  Candida auris is a yeast of mysterious origins with a mortality rate of up to 60 percent. It was identified in 2009 after it was discovered in the ear of a Japanese patient. 

                                                                        

           Microscopic image of Candida Auris

Since then, microbiologists have scoured genomic databases and natural environments for clues into its origins and evolution. Outside of people, it has only been detected in two sites: a salt marsh and a sandy beach in the Andaman Islands, a remote archipelago in the Indian Ocean.

But now it's literally become a global threat and in the U.S. too (see map). 

 Extent of Candida Auris fungus infections in U.S.

The first documented Candida auris infection in the U.S. occurred in 2016, (According to CDC researchers in a study published Monday in The Annals of Internal Medicine.)  See e.g.

Worsening Spread of Candida auris in the United States, 2019 to 2021 | Annals of Internal Medicine (acpjournals.org)

Cases of the fungus— once mostly limited to the slum areas of New York City and Chicago - have since been reported in at least 35 states and Washington, D.C. In the words of Dr. Peter Pappas, an infectious-disease specialist at the University of Alabama at Birmingham WSJ, yesterday, p. A16:

“To see a new species arrive on the scene and then suddenly emerge as a global pathogen less than 15 years later—that’s really remarkable." 

Indeed it is remarkable, especially as it's not bacterial - like plague - or a virus like Covid-19.    Most Candida auris transmission has occurred in healthcare facilities that provide long-term care to very sick patients, the ACP paper said. Select Specialty Hospital-Northwest Detroit, a rehabilitation center for the critically ill, temporarily stopped admitting patients last year after an outbreak of Candida auris. All healthcare facilities need to be aware of the threat, said Buddy Hammerman, chief medical officer of Select Medical Holdings Corp., the hospital’s operator.  Adding:

“Patients are becoming colonized with this organism and it’s becoming smarter than we are,” 

Sounds just like the fungus in 'The Last of Us', so why isn't Candida auris the feature bogey bug in that series, instead of the benign cordyceps? Maybe the producers believed it would scare too many viewers, but it surely would have made the sci-fi elements more believable.   Like the fungus featured in the HBO series Candida auris might have lived in the environment or animals for a long time before a mutation spurred its spread in people, Rising global temperatures might have encouraged Candida auris to evolve more tolerance to heat, making it more adept at infecting people, whose body temperatures have long served as an effective barrier against invasive fungal disease.  Exactly like cordyceps in the HBO series.

In a similar way to how that HBO series portrayed the origin of cordyceps (episode one)the widespread use of disinfectants and antifungals on crops might have allowed Candida auris to thrive by killing off microbial competitors.  Common disinfectants including bleach and alcohol aren’t always effective against Candida auris, which can colonize a person’s skin.  Deep cleaning and special disinfectants that destroy spores are needed to limit its spread.  According to one epidemiologist cited by the WSJ (ibid.):

“If someone is found to have Candida auris on their skin, you need to isolate them to try to prevent new people from being colonized,” 

Not everyone who encounters Candida auris is sickened, but if it infects the bloodstream, wounds or organs, the effects can be severe. The most common symptoms of Candida infection are fever and chills.  NO, there are no swollen fungus -eaten heads, such as took a chunk out of Ellie's thigh in the next to last episode of The Last of Us.  But it can lead to  candidiasis,  which means yeast invasion of mucosal surfaces inside the mouth, the respiratory tract, intestines, and vagina. (Imagine a 'yeast infection' - wherever- that can't be controlled no matter what anti-fungal or other ointment you use.

Fungal infections are estimated to kill at least 1.6 million people a year, so they are no joke.  Further, there are only three classes of antifungal drugs commonly used for severe fungal disease, all of which can cause toxic side effects. 

Worse, these rogue fungi are also developing resistance to these drugs.  At least seven Candida auris infections were resistant to available antifungals in 2021, the CDC said, up from four in the years before 2020.

Let us sincerely hope that Candida auris doesn't reach the lethal phase of becoming like the fictional version of cordyceps in The Last of Us.  Or it may very well be the last of us.

 

See Also:

• Do The Young & Unvaccinated Risk Black Fungus ...

 

Solution To Variation Of Parameters Differential Equation

 Solve using variation of parameters:

d²y/dx ²   - dy/ dx  - 2y  =  e ^3x

Solution:

This is a 2^nd order DE so write the complementary function:

y _c  = c ₁   e ^-x +  c ₂  e ^2x

Assume: 

y_p  = v ₁ e ^-x +  v ₂  e ^2x

Then:

y₁  = e ^-x ,   y ₂  = e ^2x   ,    f (x) = e ^3x

Therefore:

a) v ‘₁ e ^-x +  v’ ₂  e ^2x    = 0

b) v ‘₁  (-e ^-1) +  v’ ₂   (2e ^2x )   = e ^3x

Then we solve the above simultaneous eqns. to get:

 v’₁  = e ^-4x/3 ,   v’₂  = e ^x/3

From which we obtain:

v’₁  = e ^-4x/12                    v’₂  = e ^x/3

The procedure entails replacing the arbitrary constants  c ₁  ,  c ₂   in the complementary function by the respective functions  v ₁ and v ₂ which  will be determined.  So the resulting function:  v ₁ y ₁   +  v ₂  y ₂

will be a particular integral of:  

a _o (x) y”   +   a ₁ (x) y’  + a ₂ (x) y = b(x)

Substitute into the original eqn:

y_p  = v ₁ e ^-x +  v ₂  e ^2x

y_p  =  - e ^4x e ^-x /12  +   e ^x e^-2x/3  

=

 - e ^3x/12  + e^-3x/3  =  e^-3x/4 

Yielding the general soln.

y  =  c ₁   e ^-x +  c ₂  e ^2x +  e^-3x/4