On reaction processes with a logarithmic-diffusion

Physics Letters A 381 (2017) 94–101

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Physics Letters A www.elsevier.com/locate/pla

On reaction processes with a logarithmic-diffusion Philip Rosenau School of Mathematical Sciences, Tel-Aviv University, Tel Aviv 69978, Israel

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 14 August 2016 Received in revised form 27 October 2016 Accepted 28 October 2016 Available online 4 November 2016 Communicated by C.R. Doering Keywords: Fast diffusion Quadratic and bi-stable reactions Explicit solutions Attractors Solution’s extinction

We study formation of patterns in reaction processes with a logarithmic-diffusion: ut = (ln u )xx + R (u ). For the generic R = u (1 − u ) case the problem of travelling waves, TW, is mapped into a linear one with the propagation speed λ selected by a boundary condition, b.c. at the far away upstream. Dirichlet b.c. relaxes the process into a steady state, whereas convective b.c. u x + hu = 0, leads the system into a heating (cooling) TW for h < 1 (1 < h) or, if h = 1, into an equilibrium. We derive explicit solutions of symmetrically expanding waves and of formations which collapse in a finite time. Both are shown to be attractors of classes of initial excitations. For a bi-stable reaction R = −u (α − u )(1 − u ) we show that for α < 1/3 the system may evolve into a TW, an equilibrium, an expanding formation or to collapse. The 1/3 < α regime admits either a cooling TW or a collapse. Few other transport processes are outlined in the appendix. © 2016 Elsevier B.V. All rights reserved.

1. Introduction In the last two decades logarithmic diffusion gained remarkable notoriety due to a variety of applications in both physics and mathematics, see refs. [1,2] and references therein. An additional, not least important, argument in favor of further exploration of transport with a logarithmic diffusion is its remarkable level of solvability, a rare feature in nonlinear processes, which reveals rich and challenging phenomena. In the present work we shall unfold some of the properties due to coupling of logarithmic diffusion with reaction. In the next two sections we present the quadratic, Fisher-KPP, reaction whereas in sect. 4 we address the bistable case. In the appendix we extend the presented method to other reaction–diffusion processes. In particular we discuss formation of cavity due to radiation.

2λ

du dz

+

d2 ln u

+ u (1 − u ) = 0.

dz2

(3)

To solve Eq. (3) we introduce a map

z dζ = udz

or ζ =

u (η)dη,

(4)

−∞

under which Eq. (3) becomes linear

2. The Fisher-KPP reaction



x∈R

http://dx.doi.org/10.1016/j.physleta.2016.10.056 0375-9601/© 2016 Elsevier B.V. All rights reserved.

da dζ

(5)

,

and the b.c.: u (ζ = −∞) = 1 and u (ζ = ζ0 ) = 0. Its solution, we may assume ζ0 = 0, is

(2)

u (ζ ) = 1 − e γ ζ



Unlike the standard Fisher-KPP case wherein precursor’s dynamics is linear, the presented process is essentially nonlinear down to the

E-mail address: [email protected].

with a =

(1)

with

u (−∞) = 1 and u (+∞) = 0.



u 2λu  + u  − u + 1 = 0,

We start with the classical Fisher-KPP reaction

ut = (ln u )xx + u (1 − u ),

ground state. Moreover, if the wave’s precursor decays as u ∼ e −ax then, unlike the classical case wherein the flux −u x ∼ e −ax , here flux −u x /u ∼ a and thus remains finite as x → ∞. This has a fundamental impact on the overall dynamics and may cause a complete extinction of the process within a finite time [1]. With the classical KPP results in mind we turn to find Travelling Waves, TW, of (1)–(2). Let z = x − 2λt, then

 +

,

where λ =

1−γ2 2γ

or, since 0 < γ for the solution to stay bounded, In z coordinates we have

u ( z) =

1 1 + eγ z

.

(6)

, √

γ −1 = λ + 1 + λ2 . (7)

P. Rosenau / Physics Letters A 381 (2017) 94–101

Fig. 1. The initial kink u (x, 0) = 1/(1 + e x/2 ) converges to the steady state of Eq. (1) with h = 1. Note that everywhere but in Fig. 8 convective b.c. were used.

95

Fig. 3. The initial kink u (x, 0) = 1/(1 √ + e x/2 ) converges to a receding cooling TW of Eq. (1) with λ = −1/2 and h = (1 + 5)/2. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

This induces a cooling wave receding to the left, i.e., λ < 0. Fig. 3 displays such scenario. Alternatively, one may prescribe u at a finite distance L: u ( L ) = U 0 < 1. Now only stationary states (10) are admissible with x∗ related to U 0 and L via

x∗ = ln(

1 U0

− 1) − L .

(11)

Note the flux at x = L

−

Fig. 2. The initial kink u (x, 0) = 1/(1 + e x/4 ) converges to a heating TW with λ = 1/2 and h = 1/2. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Speed selection. So far the speed λ is arbitrary and may take both positive and negative values, with the latter corresponding to a cooling wave receding to the left. The later being a novel feature of the logarithmic diffusion. However, with the flux being finite at +∞, to assure uniqueness of the solution one has to be more specific about the action there, which is to say that one has to append the problem with an additional boundary condition upstream. The specific form of the flux in the present problem makes it natural to impose

u x + hu = 0 at x = ∞,

(8)

which uniquely relates the convection coefficient h with the propagation speed

λ=

1 − h2 2h

(9)

.

In particular, the choice h = 1 begets a steady state

u (x) =

1 1 + e x+x∗

,

x∗ = const .,

(10)

which means that when h = 1 heat production within the domain is balanced exactly by heat removed at infinity. Fig. 1 describes a typical convergence of an initial kink-like excitation into an equilibrium. Taking h < 1 reduces the amount of heat which leaves at infinity. The excess of the heat generated in the domain induces a heating wave propagating to the right and thus λ > 0 (see Fig. 2 where the chosen h = 1/2 induces a heat wave with λ = 3/4). When 1 < h more heat leaves the domain than generated within.

ux u

|x= L =

1

(12)

1 + e−L

and it attains its limiting value 1 for L >> 1. Very much as in Fig. 1, the stationary state with Dirichlet boundary condition imposed at L is an attractor: initial profiles located above (below) the equilibrium propagate to the right (left) before settling into a steady-state (not shown). 3. Expanding waves We now explore a wider family of solutions which also provides an alternative path to the TW solutions (for an application to other transport equations see the Appendix). Let v = 1/u, then in terms of v Eq. (1) reads

v t = v v xx − v 2x + ω2 (1 − v ),

x ∈ R,

(13)

where ω was added for a better trace of reaction’s impact. We seek solution in the form

v (x, t ) = A (t ) + B (t ) f (x)

(14)

which satisfies

˙ − ω2 (1 − A ) = −( B˙ + ω2 B ) f + A B f  + B 2 [ Q ( f )] A . and Q ( f ) = f f  − f  2 . We constrain Q ( f ) to satisfy

(15)

f f  − f  2 = α0 + α1 f ,

(16)

α0 , α1 consts.

Further developments depend on

α0 and α1 .

1) Let α0 = α1 = 0, then f (x) = exp(γ x). Solving for A and B we find for both γ and the solution an ω = 1 extension of Eqs. (6) and (7). 2) Let

u (x, t ) =

α0 = 0 and α1 = −2. Then f (x) = x2 and thus 1 A (t ) + B (t )x2

with A (t ) found solving

,

(17)

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P. Rosenau / Physics Letters A 381 (2017) 94–101

˙ + ω 2 ( A − 1) = 2 A B , A

B=

ω2 2(e τ − 1)

and τ = ω2 t. The explicit form of A is a bit cumbersome, but for 1 << τ

A∼ = 1 + e −τ [ A 0 + τ ], whereas for

u=

τ << 1; A ∼ τ ( A 1 + ln τ ). For ω = 0 it reduces to

2t 2 A 0 t 2 + x2

which is a response to a source located at the origin, [1]. A spherical extension of case (2) is outlined in the Appendix. 3) Let α1 = 0 and α0 = 0. ⇒ f (x) = cosh(γ x) with γ 2 = α0 (if α0 < 0 then f (x) = cos(γ x) with γ 2 = |α0 |). A(t) and B(t) are found via

˙ − ω2 (1 − A ) = γ 2 B 2 A

and B˙ + ω2 B = γ 2 A B .

(18)

Though, unless ω = 0, the general solution for A and B is yet to be found, its essential features are easily accessed; A) Time independent solutions. Let

u eq (κ ) =

Fig. 4. Evolution of u (x, 0) = 3u eq (x; κ = 0.9). At first the relatively large excitation decays under the critical u = 1 threshold whereas its core is hardly moving. Thereafter it evolves into a 2-sided expanding wave comprised of two interconnected kink-like fronts approaching a constant speed. h = ±0.9 at x = ±20.

.

κ = γ /ω ≤ 1, then

κ2 , κ < 1. 1 + 1 − κ 2 cosh(γ x) √

(19)

(κ = 1 begets the steady kink

u=

1 1 + eωx

which is a subcase of (7).) It is easy to see that these equilibria are unstable. Perturbing Eqs. (18) around A = κ −2 and B 2 = κ −2 − 1, yields the respective eigenvalues

2δ± = −1 ±



1 + 8(1 − κ 2 ).

The equilibrium is thus a saddle point of Eqs. (18) and perturbation’s trajectories run away. B) Bounded states. As A → 1 and B → 0 we obtain





2 B ∼ e −(1−κ )τ B 0 + κ A 0 B 1 e −τ + ...

and ( A 0 , B 0 are consts.)

A ∼ 1 + e −τ T (τ ) + ...





where T (τ ) = A 0 + B 20 κ 2 a(τ ) ,

with

a(τ ) =

2 eκ τ

2κ 2 − 1

and a(τ ) = τ for

u=

for

κ 2 = 1/2

κ 2 = 1/2. Thus, 1

2 1 + e −τ T (τ ) + B 0 e −(1−κ )τ cosh(γ x)

(20)

which describes a wave which expands symmetrically. At large |x| the fronts look like two kinks traversing in opposite directions with their speed converging to a constant value. For 0 << x, u ∼ e −γ (x−2λt ) , whereas for x << 0; u ∼ e γ (x+2λt ) : a wave steadily receding to the left. Figs. 4–5 display emergence of such solutions out of an initial excitation. In particular, note the logarithmic display in the lower plate of Fig. 5 which reveals kink-like shape of fronts with their expansion speed approaching a constant value whereas their interconnection ascends to the limiting u = 1 state. C) Collapsing states. A ∼ = B → ∞. Here contribution of the

ω-terms due to reaction is secondary. To a leading order

Fig. 5. Upper plate: evolution of u (x, 0) = 2u eq (x; κ = 0.5) into a 2-sided expanding wave formation comprised of two interconnected kink-like fronts approaching a constant speed. Lower plate: the logarithmic scale of 1 − u clearly indicates the very quick ascent of formation’s top toward u = 1 and the kink-like shape of its fronts. h = ±0.5 at x = ±40. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

A (t ) = A 0 coth( A 0 γ 2 tˆ),

B (t ) =

A0 sinh( A 0 γ 2 tˆ)

where tˆ = t 0 − t and thus λ = A 0 γ ,

,

(21)

P. Rosenau / Physics Letters A 381 (2017) 94–101

u=

γ

97



sinh(λγ tˆ)

λ cosh(λγ tˆ) + cosh(γ x)

(22)

.

This solution collapses within finite time. It may be also rewritten as

u=

γ 

2λ

tanh[

γ 2

γ (x − λtˆ)] − tanh[ (x + λtˆ)] 2

(23)

and describes motion of two receding kinks. Since for large A and B, u << 1, we may approximate A ∼ =B∼ = γ 2 /tˆ and thus

u=

γ 2tˆ , t ≤ t0 . 2 cosh2 (γ x/2)

(24)

The approximations leading to the collapsing solutions correspond to an essentially pure diffusion with ω = 0 (in fact both solutions were thus derived in [1] using a different approach). One way to tie the collapsing states with the evolving ones is to approximate the reaction for small u and apply the map

v = e −ω t u and 2

τ=

1

ω2

[1 − e −ω t ], 2

Fig. 6. Initial excitation located under the steady state, u (x, 0) = 0.9u eq (x; κ = 0.9), quenches within a finite time. h = ±0.9 at x = ±20. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

which renders a purely diffusive process: v τ = (ln v )xx . In terms of the original variables, now (24) reads 2

γ 2 [1 + a∗ eω t ] . u= where a∗ = t 0 ω2 − 1. 2ω2 cosh2 (γ x/2)

(25)

Alternatively, noting that for u << 1 the solution is separable u = φ(t ) v (x) we obtain (25). Evolution of (25) depends crucially on the sign of a∗ with a∗ = 0 being an equilibrium of the simplified problem

u eq (κ ) =

κ2 , κ 2 << 1, 2 cosh2 (γ x/2)

(26)

which separates between lasting and collapsing states (it is, of course, the κ << 1 limit of Eq. (19)). Now assume that u (x, t = 0) = u 0 cosh−2 (γ x/2). Then if κ 2 /2 < u 0 , the solution grows exponentially until reaction’s quadratic term tempers its growth, see Figs. 4–5, and evolves it into a travelling formation. In contradistinction, if u 0 < κ 2 /2 ⇒ a∗ < 0 the solution quenches at

t∗ = −

ln(1 − ωt 0 )

ω2

Fig. 7. At first the super-critical convection with h = ±1.2 at x = ±20, causes the initial excitation to shrink and its amplitude to grow. This phase last for a while, then the process starts collapsing and terminates within a finite time. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

.

Fig. 6 describes a collapsing state which could have been also inferred from a comparison theorem: when an initial excitation is below the equilibrium (26) it quenches within a finite time. Evolution and collapse. Summing our results, we first assume convective boundary conditions

u x ± hu = 0 at x = ±∞,

(27)

with equilibria being admissible for any κ = h < 1. The emerging patterns may be then classified as follows. A) Let h < 1. If the initial excitation is above the equilibrium (19), i.e. 0 ≤ u 0 (x, 0) − u eq (x; h), as exemplified by Figs. 4–5, the process will evolve into an expanding wave formation (22). But if u (x, 0) − u eq (x; h) ≤ 0 then the process, as exemplified in Fig. 6, quenches within finite time. B) Let 1 < h. The super-critical convection κ = h > 1 (ω = 1) causes any initial excitation, as exemplified by Fig. 7, to quench within a finite time. To appreciate the vital role of boundary conditions compare Figs. 5 and 8 where the Dirichlet b.c.

Fig. 8. Initial excitation as in Fig. 5. Though at first evolution in both cases follows the same pattern, the Dirichlet b.c. at x = ±40 force the wave to settle into a steady state. Location of its sharp transition fronts is determined by u ( L ). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

98

P. Rosenau / Physics Letters A 381 (2017) 94–101

u (±) = U 0 < 1, are used. As before, Eq. (19) is an equilibrium, but now γ is determined via U 0 . To determine it note that since 1 << L we may assume that κ = 1 − 2 , << 1. To a leading order we thus obtain

2

1 U0

 − 1 e−L ,

which represents an almost flat equilibrium with a very fast transition boundary layer at each edge. Thus, unlike the convective case, the process, as exemplified in Fig. 8, evolves toward an equilibrium with the prescribed value of U 0 at x = ± L. 4. Bistable cubic reaction 4.1. A linear diffusion We start our bistable affairs with the standard linear diffusion augmented with a nonlinear convection

ut + 2β uu x = u xx − u (α − u )(1 − u ),

xε R,

(28)

together with the b.c. (2). To find its TW, let z = x − λt:

−λ

du dz

+β

du 2 dz

=

d2 u dz2

− u (α − u )(1 − u ).

(29)

Applying map (4) turns the problem into a quadratic entity

−λu  + β(u 2 ) = (uu  ) − (α − u )(1 − u ),

1

1 − 2α

γ = {β + 2 + β 2 }, and λ = β +  2

2 + β2

.

(30)

Thus, the given α and β impose a unique speed of propagation. Notably, though the profile is independent of α , it strongly depends on β (actually, if you assume in (29) v = 1/u, and in the resulting ODE take v = 1 + e γ z , both the profile and its speed follow). Note that for each α there is a critical convection βc such that at

βc =

2α − 1 2

√

α

,

α=

√

1 4

reaction with a logarithmic diffusion and critical convection

h∗ (1/4) = 1/2 3, see Eq. (35), converges to a stationary kink.

u 2x − u 2 P 2 (u ; α ) = E 0 ,

P 2 (u ; α ) = α − u ∗ u +

u2 2

,

(32)

where u ∗ =2(1 + α )/3. For 0 < α < 1/2, P 2 has two distinct roots; u ± = u ∗ ± u 2∗ − 2α . For the sought after solution we take E 0 = 0 and u ∈ [0, u − ] and integrate Eq. (32). For α = 1/2 the two roots u ± coalesce and the solitary formation becomes a kink. 4.2. A logarithmic diffusion

for which, in analogy with [3] one finds (7) to be the sought after solution with



Fig. 9. Bistable

(31)

the motion comes to a halt. Further increase in |β| reverses its direction turning a heating (cooling) wave propagating to the right (left) into a cooling (heating) wave propagating to the left (right). We pause to re-examine the α = 0 case (with β = 0) wherein R = ω2 u 2 (1 − u ) ≥ 0. It calls for a comparison with the classical quadratic KPP case, as in both cases R (u ) ≥ 0. As the total heat

1 production 0 R (u )du = 1/6 for KPP and ω2 /12 for the present case, respectively, to equalize the √ total heat production in both cases we set ω2 = 2. ⇒ λ = ω/ 2 = 1, yielding half of the λ = 2 speed in the KPP case. Another noteworthy case is α = −1;

ut = u xx + ω2 u (1 − u 2 ).

1 Now 0 R (u )du = 1/4 and we pick ω2 = 2/3 to equalize the to√ tal heat with the KPP case. Using (30) yields λ = 3  1.73 which is closer to the KPP speed than the α = 0 case. It thus appears that front’s speed is far more affected by the heat available at the precursor than by its global amount. Finally, we note that the stationary kink obtained for α = 1/2 in (30) is a limiting case of a wider family of stationary solitary formations available for 0 < α ≤ 1/2. To this end let β = 0. Integrate (28) once to obtain

We proceed with logarithmic diffusion augmented with convective boundary conditions (8)

ut = (ln u )xx − u (α − u )(1 − u ),

x ∈ R.

(33)

Based on the insight gained we seek stationary solutions of Eq. (33). Integrating once we cast it into

3

 u 2 x

u

  − (1 − u )2 (2u + R (1)) − e 0 = 0,

(34)

where R (1) = 1 − 3α and e 0 is a constant of integration chosen to assure that it vanishes with u x at u = 1. Kinks. We address first the resulting kink and note the finite flux at u = 0

−u x /u = h∗ (α ) =

 (1 − 3α )/3.

(35)

Clearly, such kinks are possible as long as α < 1/3 and as numerical experiments reveal, are strong attractors of kink-like excitations. This is clearly seen in the example in Fig. 9. When h < h∗ (α ) then, as seen in Fig. 10, a heating TW emerges. Similarly, h∗ (α ) < h leads to an excessive heat extraction at infinity which causes a cooling travelling wave receding to the left (not shown). The α = 1/3 case is exceptional. Now the flux vanishes at u = 0. Solving (34)

2

±√ x = 3

−2 v

+ ln

1 + v  1−v

,

where u = v 2

and u ∼ 1/x2 as |x| → ∞. This is an algebraic decay and the flux ∼ −2/x. As in the conventional case, no heat flux escapes the system. For 1/3 < α , R (1) < 0. Then the net absorption of heat induces a cooling TW receding to the left (not shown). Solitary formations. Inspection of (34) reveals that for stationary solutions to exist we need α ≤ 1/3 and 0 < e 0 < 1 − 3α . At u = 0 we have the flux

−u x /u = h∗ (α ) =

 [1 − (3α + e 0 )]/3,

(36)

P. Rosenau / Physics Letters A 381 (2017) 94–101

Fig. 10. Bistable α = 14 reaction with a logarithmic diffusion and subcritical convection h = 0.1, see Eq. (35), evolves into a travelling kink.

√

Fig. 11. Bistable α = 14 reaction, u (x, 0) = 1.2u eq (x) and h = ±1/ 24, evolves into an expanding formation with each front’s velocity approaching the speed of the respective TW. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

which for a given α < 1/3 relates h with e 0 ; there is an admissible band of e 0 and a corresponding domain of admissible convection coefficients h which admit stationary formations (this is to be contrasted with the conventional case where for each admissible α ≤ 1/2 there is only one steady state). Otherwise there is no equilibrium. Given convective b.c. (27), we then have the following characterization of the dynamics A:

α < 1/3. √

a1: h < (1 − 3α )/3. The system admits a 1-parameter family equilibria u eq (x; α , h) which, as in the quadratic case, are unstable: initial excitation above u eq (x; α , h) evolves into an expanding formation, see example in Fig. 11, whereas excitation below equilibrium quenches within finite time (not shown). a2:

√

(1 − 3α )/3 < h: All solitary initial excitations quench.

99

Fig. 12. Bistable α = 12 reaction with initial excitation as in Fig. 11. Here the process quenches within a finite time. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

the reaction–diffusion spiel; the finite flux at infinity gives to the faraway action a much larger impact on the overall dynamics than encountered in the standard processes. The system behaves as though the action is taking place in a finite domain. The convective boundary conditions select a unique speed of propagation which may correspond to a heating wave, a cooling wave or an equilibrium which is an explicit statement of balance between heat production within the domain and the heat extracted at the boundaries. The later is the only outcome if Dirichlet b.c. are imposed. The interplay between the heat generated in the domain and removed at the boundaries begets a rich variety of patterns, with the eventuality of extinction within a finite time being a unique feature of the fast-diffusion. In the Appendix a wider class of transport equations was outlined. In the case of a simple plasma thermal model it reveals collapsing thermal states due to the Bremsstrahlung radiation and formation of an expanding cavity. Finally, though on the scientific level we have unfolded fairly well the discussed processes, the actual hard core mathematical proof of the attractive nature of the presented solutions, is yet to be provided. Unlike the slow diffusion case presented in [3], no shortcut to this effect seems possible. Acknowledgements I sincerely thank Mr. A. Zilburg for his help with the numerical simulations presented. This work was supported in part by the Bauer-Neuman Chair in Applied Mathematics and Theoretical Mechanics. Appendix A. Related problems We extend the reach of the approach used in sections 2 and 3 to a wider class of nonlinear transport equations, with the Lagrange map applied to



ut = un u x

 x

+ b1−n u 1−n + b−n u −n ,

x ∈ R,

(A.1)

B: 1/3 < α : All solitary initial excitations quench. Fig. 12 exemplifies such a case.

whereas the approach of sec. 3, to

5. Closing comments

ut = un u x

The super fast logarithmic diffusion applied in this work to both the quadratic and bistable reactions fundamentally changes

Both equations have a cross over at n = −1/2 which will be utilized shortly.



 x

+ b1−n u 1−n + b1 u + b1+n u 1+n .

(A.2)

100

P. Rosenau / Physics Letters A 381 (2017) 94–101

We start with Eq. (A.1) and define a map

dζ = u

−n

z dz

dη

ζ=

or

un (η)

−∞

(A.3)

.

Applied to the TW rendition of (A.1), z = x − 2λt, it begets a linear entity:

2λu  + u  + b1−n u + b−n = 0,

with a =

da dζ

.

(A.4)

In what follows for the n = −1/2 case in (A.1) we shall assume b3/2 = 1 and b1/2 = −1. The chosen diffusion coefficient corresponds to the Okuda–Dawson scaling [4]. Thus

ut = (u −1/2 u x )x + u 3/2 − u 1/2 ,

xε R,

(A.5)

with the last term corresponding to the Bremsstrahlung radiation in plasma. Its TW solution



u (ζ ) = 1 − e −|γ± ζ |

 +

and



γ ± = λ ± λ2 − 1 ,

(A.6)

in z coordinates reads

u (x, t ) = tanh2

z

(A.7)

.

2

However, in spite of the sub-linear nature of the absorption term, the particular form of the fast diffusion precludes sharp fronts. The solution (A.7) has to be thus understood as a travelling cavity. Yet, as we shall shortly see, this is far from end of the story. Turning to Eq. (A.2) we express it in terms of v = un



v t = v v xx +

1 n



v 2x + l1−n + l1 v + l1+n v 2 ,

(A.8)

where lk = nbk . Applying Ansatz (14)





˙ − l1−n + l1 A + l1+n A 2 − B 2 { Q } = A

  = − B˙ f + A f  + (l1 + 2l1+n A ) f B ,

(A.9)

where { Q } = f f  + n1 f  2 + l1+n f 2 . The last term in (A.9) calls for f  = κ f , κ = const . or f  = const . For the later case to be consistent with { Q } we need to set l1+n = 0 as well, which begets a quadratic entity in x. This enables to extend our results to a spherical case



n

ut = u ur

 r

+

Nun r

u r + b1−n u

1−n

+ b1 u ,

(A.10)

where N = 1, 2, 3, with solutions of the form



u = A (t ) + B (t )r 2

 n1



˙ − l1−n + l1 A = 2 A B (1 + N ), A B˙ = 2( N + 1 +

2 n

) B 2 + l1 B .

(A.11)

For n = −1 this is an axisymmetric extension of the logarithmic case wherein b1 = 1 and b2 = −1. Note that in the planar N = 1 case, B is determined via a linear equation B˙ = −b1 B and A → 1 for a large time. To apply the f  = κ f constraint assuming κ > 0 implies f = e γ x or cosh(γ x) with κ = γ 2 . In the first case { Q } has to vanish which determines γ

nl1+n + (1 + n)γ 2 = 0.

For n = −1 it leaves γ unconstrained (with b0 = 0). Otherwise, γ 2 = −n2 b1+n /(1 + n) provided that b1+n < 0 is assumed. A and B are then determined via

˙ = l1−n + l1 A + l1+n A 2 , B˙ − l1 B = A

(A.12)

2 + n 1+n

l1+n A B .

(A.13)

Finally, let f = cosh(γ x). Now { Q } = −γ 2 /n. B is determined via (A.13) whereas A via, n = −1,

˙ = l1−n + l1 A + l1+n A 2 + A

.

A and B are determined via



Fig. 13. The TW solution (A.7) given at t = 0 morphs at once into an expanding cavity. The almost equidistance of cavity’s walls at the upper plate indicates a constant speed of expansion. The essentially constant profiles in logarithmic coordinates in the lower plate imply exponential profile everywhere but at the edges of the cavity. Note the sinking of cavity’s bottom toward the ground state. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

l1+n 1+n

B2.

(A.14)

An example. Two pairs of ±n are of particular interest: n = ±1 and n = ±1/2. The n = 1 case was addressed in part in [3], see also [5] and references therein, whereas the n = −1 case in the main body of the paper. The other pair is



u t = u ± 0 .5 u x

 x

+ b 1 ∓ 0 .5 u 1 ∓ 0 .5 + b 1 u + b 1 ± 0 .5 u 1 ± 0 .5 .

In v coordinates

vt =

v v xx ± 2v 2x

+

(A.15)



l1/2 + l1 v + l3/2 v 2 ; n = +1/2 l3/2 + l1 v + l1/2 v 2 ; n = −1/2.

(A.16)

The choice of f = x2 calls for b1±1/2 = 0 with n = ±1/2, respectively. Otherwise (A.12) implies b1±1/2 < 0. We thus set b1±1/2 =

√ √ −1 ⇒ γ = 1/ 6 and 1/ 2, for n = ±1/2, respectively.

P. Rosenau / Physics Letters A 381 (2017) 94–101

u eq (x) =

101

2 cosh2 ( √x ) 2

(A.19)

, √

corresponds to A = 0 and B = 1/ 2. A collapsing, solution follows assuming large A and B. Neglecting the constant in (A.18) and solving for A and B, yields A 2 = B 2 + C ∗ B 2/3 . Thus A ∼ B = 2/3(t 0 − t ). At a later stage of collapse the solution then takes the form

u=

9(t 0 − t )2 x 16 cosh4 ( √ )

.

(A.20)

2 2

With the neglect of the source u 3/2 , (A.20) becomes an exact solution of Eq. (A.5).

Fig. 14. The wedged domain marks a region in space where 0.1 < u (x, t ). The straight slope of its boundaries  ±1.88 reveals that the speed of cavity edges approaches from below the lowest speed value λ = ±1 of its TW.

For the n = −1/2 case we return to Eq. (A.5) and assume

u= 

1

2 ,

A (t ) + B (t ) cosh( √x )

(A.17)

References

(A.18)

[1] [2] [3] [4] [5]

2

where

B˙ =

3 2

A B,

˙+ and A

1 2

(1 − A 2 ) = B 2 .

First, note that the equilibrium solution of (A.4)

Formation of cavity. The TW solution (A.7) suggests a travelling cavity, however its numerical simulation, see Fig. 13, reveals a very different scenario; the TW morphs at once into a two sided expanding cavity with its form being almost a mirror image of the expanding formations in Figs. 4–5, with its fronts, see Fig. 14, approaching very quickly from below a constant velocity coinciding with the lowest admissible velocity λ = ±1 of its TW (A.6).

P. Rosenau, Phys. Rev. Lett. 74 (1995) 1057. P.P. Daskalopoulos, P. Hamilton, N. Sesum, J. Differ. Geom. 91 (2012) 171. P. Rosenau, Phys. Rev. Lett. 88 (2002) 194501. P.H. Okuda, J.M. Dawson, Phys. Fluids 16 (1973) 408. J.D. Murray, Mathematical Biology, Springer, 2002.