The shocks in Josephson transmission line revisited

We continue our previous studies of the localized travelling waves, more specifically, of the shocks and the kinks, propagating in the series-connected Josephson transmission line (JTL). The paper consists of two parts. In the first part we calculate the scattering of the"sound' (small amplitude small wave vector harmonic wave) on the shock wave. In the second part we study the similarities and the dissimilarities between the shocks and the kinks in the lossy JTL. We also find the particular cases, when the nonlinear equation, describing weak travelling wave in the lossy JTL can be integrated in terms of elementary functions.


I. INTRODUCTION
The interest in studies of nonlinear electrical transmission lines, in particular of lossy nonlinear transmission lines, has started some time ago [1][2][3] , but it became even more pronounced recently [4][5][6][7] .A very recent and complete review of studies of nonlinear electric transmission networks one can find in Ref. 8 .
We studied previously the shock waves in the lossy Josephson transmission line (JTL) JTL 9,10 and kinks (and solitons) in the lossless (actually, without any shunting at al) JTL 10 .The present work had several aims.First we would like to analyse the interaction between the "sound" (small amplitude small wave vector harmonic wave) and the shock wave.Second we would like to establish the relation between the shock waves and the kinks.And third, we would like to additionally study the weak shock waves, and, in particular, to look for the cases when the nonlinear equation, describing weak travelling waves in the JTL, can be integrated analytically.
The rest of the article is constructed as follows.In Section II we rederive the circuit equations describing the JTL in the continuum approximation.In Section III we consider scattering of the "sound" wave by the shock wave and calculate the appropriate reflection and transmission coefficients.In Section IV we show that the kinks, which we previously believed to exist only in the lossless JTL, exist also in the lossy JTL and show the connection between the shocks and the kinks.We also analytically integrate the wave equation describing weak shocks in the lossy JTL for the specific value of the losses parameter.We conclude in Section V.In the Appendix A we present a physically appealing model of the JTL, composed of superconducting grains.In the Appendix B we explain the condition for the applicability of the continuum approximation used in the paper.son junctions (JJs) capacitors and resistors is shown on Fig. 1. (Possible physical realization of the model is presented in the Appendix A.) We take as the dynamical variables the phase differences (which we for brevity will call just phases) φ n across the JJs and the voltages v n of the ground capacitors.The circuit equations are ℏ 2e where C is the capacitance, I c is the critical current of the JJ, and C J and R J are the capacitor and the ohmic resistor shunting the JJ.In the continuum approximation we treat n as the continuous variable Z and approximate the finite differences in the r.h.s. of the equations by the first derivatives with respect to Z, after which the equations take the form where we introduced the dimensionless time τ = t/ √ L J C and the dimensionless voltage V = v/(Z J I c ); L J ≡ arXiv:2307.15078v1[cond-mat.supr-con]18 Jul 2023 ℏ/(2eI c ) is the "inductance" of the JJ and Z J ≡ L J /C is the "characteristic impedance" of the JTL.The condition for the applicability of the continuum approximation is formulated explicitly in the Appendix B.

III. THE SOUND SCATTERING BY THE SHOCK WAVE
A. The sound waves and the shock waves Equation (2b) being nonlinear, the system (2a), (2b) has a lot of different types of solutions.In this Section we'll be interested in only two types of those.First type -small amplitude small wave vector harmonic waves on a homogeneous background φ 0 .For such waves Eq. ( 2b) is simplified to We ignored the shunting terms in r.h.s. of (2a) because they contain higher order derivatives in comparison with the main term, and small wave vector means also small frequency.
The harmonic wave solutions of Eqs.(2a), (2b) (which, for brevity we'll call the sound) are where is the normalized sound velocity.In this paper the normalized velocity ≡ physical velocity times √ L J C/Λ, where Λ is the JTL period.Note that the stability of a homogeneous background φ 0 demands cos φ 0 > 0. ( The second type of solutions we'll be (mostly) interested in, is shock waves 9,10 .In this Section we'll ignore the structure of the shock wave and consider it as the discontinuity of the dynamical variables.The property of the shocks, which will be proven in the next Section, connects the discontinuities of φ and V with the shock velocity: where φ 1 and V 1 are the phase and the voltage before the shock, φ 2 and V 2 -after the shock, and U is the normalized shock wave velocity.Note also the obvious result of (7a), (7b):

B. The reflection and the transmission coefficients
In this Section we'll be interested in two problems 11 .The first one: A sound wave is incident from the rear on a shock wave.Determine the sound reflection coefficient.The situation is shown in Fig. 2 we took into account the equation, which will be derived in Section IV where φ b and φ a are the phases before and after the shock in the absence of the sound respectively.Also, For the first problem mentioned above we have where (in) stands for the incident sound wave and (r) for the reflected sound wave.Substituting ( 10) -(11d) into (7a), (7b) in the first order approximation we obtain Taking into account the relations (the difference in the signs is because of the opposite directions of propagation of the two waves) and excluding δU we obtain where u in = u (φ a ) − U is the velocity of the incident sound wave relative to the shock wave, and u r = u (φ a ) + U is the velocity of the reflected sound wave relative to the shock wave.As one could have expected, the modulus of the sound reflection coefficient is less than one, and it goes to zero when the intensity of the shock wave decreases, that is when φ a → φ b , in other words, when the shock wave itself nearly becomes the sound wave.Now let us turn to the second problem.We have where (t) stands for the transmitted wave.Substituting ( 10), (15a) -(15d) into (7a), (7b), in the first order approximation we obtain Taking into account the relations and excluding δU we obtain where u in = u (φ b ) + U is the velocity of the incident sound wave relative to the shock wave, and U is the velocity of the transmitted sound wave relative to the shock wave.As one could have expected, the sound transmission coefficient is less than one, and goes to one when the intensity of the shock wave decreases, that is when φ a → φ b .Looking back at the derivation of ( 14) and ( 18) we understand that the equations will be valid also for a generalized Josephson law for the supercurrent I s : where f is a (nearly) arbitrary function.The difference from the case considered above is that the sound velocity in the general case is and the shock velocity is given by the equation The validity of (21) will become obvious after we present the proof of its particular case (8) in the next Section.

A. The travelling waves
In this Section we would like to study the structure of the shock wave, so we return to Eqs. (2a), (2b) in their full glory.For the travelling waves, where U is the travelling wave velocity.Making the ansatz we obtain Consider a solution which for τ ∈ (−∞, +∞) stays in the finite region of (φ, V ) phase space.The limit cycles are excluded for our problem, and strange attractors are excluded in a 2d phase space in general 13 .Hence the trajectory begins in a fixed point and ends in a fixed point Integrating (23a), (23b) with respect to τ from −∞ to +∞ and taking into account the boundary conditions, we obtain Eqs.(7a), (7b), which are the basis of our consideration in the previous Section.Note that the shunting of the JJ doesn't influence the shock velocity 9,10 .
Excluding V from (23a), (23b) and integrating the resulting equation we obtain closed equation for φ where τ ≡ τ C/C J = t/ √ L J C J , γ ≡ L J /C J R J and F is the constant of integration.Taking into account the boundary conditions (24a), we can write down (25) as where which reminds the equation describing current biased JJ within the RCSJ model 12 .
The potential (30) should have maximum at φ 1 , while the point φ 2 should be a stationary point of the potential with the property from which follows −φ 1 < φ 2 < φ 1 .The point φ 2 can be either a minimum or a maximum.The boundary between these two cases (when φ 2 is an inflexion point) we can find by equating the second derivative of the potential at the point φ 2 to zero The approximate solution of (32) is φ 2 = −φ 1 /2.What was said above can be reformulated in a slightly different way.Because the physics is obviously symmetric with respect to simultaneous inversion of all phases φ → −φ, in the following we consider only φ 1 ∈ (0, π/2).If φ 2 is positive, it is inevitably the point of a minimum of the potential.In fact, the stationary points of the potential are given by the equation Because sin φ is concave downward for 0 < φ < π/2, the straight line, crossing the sine curve at the points π/2 > φ 1 , φ 2 > 0 can't cross the curve in between.Hence there are no stationary points between φ 1 and φ 2 .The potential Π(φ) for positive φ 2 is illustrated in Fig. 4.
On the other hand, for φ 2 < 0 the potential Π(φ) can have either a minimum or a maximum at φ 2 , as it is illustrated in Fig. 5.
Looking at Fig. 5 (left) we realize that for the solution with φ 1 and φ 2 having opposite signs to exist, the effective friction coefficient γ should be large enough to prevent escape of the particle above the potential barrier to the left of φ 2 .(There is no such restriction for the shock wave with φ 1 and φ 2 having the same sign, because in this case the left potential barrier is higher than the right one, as it is illustrated in Fig. 4.) The minimum of the potential at φ 2 situation corresponds to the shock wave and was discussed at length in our previous publications 9,10 .Equation ( 29) can be easily integrated numerically.The result of such integration is presented in Fig. 6.The maximum situation we considered previously only for the particular case of the JTL in the absence of shunting 10 .We called such travelling waves the kinks.Now we understand that similar kinks exist also in the lossy JTL (for −φ 1 < φ 2 < φ 1 /2).Looking at Fig. 5 (right), presenting the potential for the kink, we realize, that since the particle stops at the unstable equilibrium point, for kink to exist, the fine tuning of γ is necessary.Saying it in different words, for a given φ 1 and given γ, only the kink with the definite value of φ 2 can exist.In particular, in the absence of losses (γ = 0) only the kinks with φ 2 = −φ 1 , are possible 10 .
Everywhere above we considered the travelling wave going to the right, but, of course, by interchanging φ 1 and φ 2 we obtain the wave going to the left.So the conditions for the shocks and for the kinks in the whole phase plane of the boundary conditions (φ 1 , φ 2 ) are shown in Fig. 7. Two additional straight lines on this Figure φ 2 = −φ 1 and φ 2 = φ 1 present the kinks and the solitons respectively, which can exist in the bare-bones (unshunted) JTL 10 and propagate in both directions.

C. The shocks velocity vs. the kinks velocity
Differentiating the r.h.s. of (25) with respect to φ we obtain For the shock φ 1 is the point of a maximum of Π(φ) and φ 2 is the point of a minimum.Hence the second derivative of the potential with respect to φ is negative at φ 1 and positive at φ 2 .Thus The inequalities (35) reflect the well-known in the nonlinear waves theory fact: the shock velocity is smaller The phase plane of the boundary conditions (φ1, φ2).Blue regions correspond to the shock wave moving to the right, green regions -to the left.Yellow regions correspond to the kink moving to the right, red regions -to the left.The thick black line φ2 = −φ1 corresponds to the kink, the thick black line φ2 = φ1 -to the soliton which can exist only in the bare-bones JTL and propagate in both directions.
than the sound velocity in the region behind the shock but larger than the sound velocity in the region before the shock 14 .
From the inequalities (35) we can prove that a shock can not split into two shocks.Actually we can make even stronger statement: two shocks moving in the same direction will merge.In fact, let there is the first shock φ 2 ← φ 3 and the second shock φ 3 ← φ 1 ahead of it.Because of inequalities (35) the velocity of the first shock is larger, and the velocity of the second shock is smaller that u(φ 3 ).The statement is proved.Note that due to a onedimensional nature of our problem we don't have to consider the corrugation instability of the shock wave [15][16][17][18][19] .
For the kink both φ 1 and φ 2 are the points of minima.Hence the second derivative of the potential with respect to φ is positive at both points.Thus The kink is supersonic from the point of view both of the region before and after it.

D. Weak shock waves
For weak wave, characterized by the condition |φ 1 − φ 2 | ≪ 1, the r.h.s. of (25) can be approximated as where As the result, (25) can be simplified to Let us make the change of independent variable where the parameter β will be chosen later.After the change of variable, (39) takes the form We are looking in this Subsection for analytical solutions of (39).After the change of variable, one such solution 20 (existing for the appropriate relation between α and γ which will be determined immediately) can be found by inspection: Actually, there are similar solutions for other two pares of indices, but in the last moment we have recalled the boundary conditions (24a), which after the change of the independent variable turn into Chosen by us solution, in distinction from other two, satisfies the boundary conditions (43).Substituting (42) into (41) we obtain We realize that equation (44) turns into identity, provided β and γ satisfy the relations Solving (45a), (45b) we obtain So finally, if γ satisfies the condition (46b), the solution of (39) with the boundary conditions (24a) is Equations ( 46b) and (47) are applicable both to the weak shocks and to the weak kinks.In particular, for φ 2 = −φ 1 the equations give γ = 0 and Let us return to Eq. ( 39) and strengthen the assumption which lead to the latter to |φ 1 − φ 2 | ≪ |φ|.In this case the equation can be approximated as where α ′ ≡ sin φ/2, and Eq.(41) takes the form Again a solution can be found by inspection: Substituting ( 51) into (50) we obtain We realize that equation (52) turns into identity, provided β and γ satisfy the relations Solving (53a), (53b) we obtain So finally, if γ satisfies the condition (54b), the solution of (49) with the boundary conditions (24a) is Pay attention that though (49) is an approximation to (39), it can be integrated analytically for totally different value of γ (and hence the analytic solutions (55) and (47) are totally different).

V. CONCLUSIONS
The interaction of the sound waves with the shock waves is well studied in fluid mechanics.In Section III we considered similar problem for the JTL.The formulas for the reflection coefficient in one case and the transmission coefficient in the other case (Eqs.( 14) and ( 18)) turned out to be very simple and appealing.
We established the relation between the shocks existing in the lossy JTL and the kinks, which as we now understand, exist both in the lossy and in the lossless JTL.However the solitons, we studied previously in the lossless JTL, are absent in the lossy JTL.
We found the particular cases when nonlinear equation describing weak travelling waves in the lossy JTL can be integrated analytically.A physically appealing model of the JTL composed of superconducting grains is presented in Fig. 8. (For simplicity in this Appendix we ignore the shunting capacitor.)Here, we take as the dynamical variables the We realise that Eqs.(1a), (1b) (in the absence of the shunting capacitor) follows from Eqs. (A1a), (A1b) if we substitute φ n = Φ n−1 − Φ n .Also, if we exclude v n from (A1a), (A1b) we obtain which is the particular case of the Fermi-Pasta-Ulam-Tsingou equation (with losses).It is interesting to compare (A2) with the equation from Ref. 21 , describing the chain of interacting particles with friction where m is the mass and y n are displacements of particles in the chain, U (z) is the potential of the interparticle interaction, and α is the friction coefficient.Comparison shows the substantially different character of the losses in the systems.
It is also interesting to compare the JTL with the one-dimensional Josephson-junction array.The equation describing the fluxon dynamics in the array is the discretized version of the perturbed sine-Gordon equation 22 where α is the dissipation coefficient.It is appropriate to compare (A4) with the equation obtained by excluding v n from (1a), (1b) Again, the comparison shows the substantially different character of the losses in the systems.But even in the absence of losses (A5) is different from the sine-Gordon equation.Neither does (A5) in the continuum approximation coincides with the sine-Gordon equation with losses 23 .

Appendix B: The continuum approximation
Natural question is how good is the continuum approximation used everywhere in this paper?To answer this question let us return to Eqs. (1a), (1b) and exclude v n .We obtain The continuum approximation (in the narrow sense) consists in promoting the discrete variable n to the continuous variable Z and approximating the discrete second order derivatives in the r.h.s. of (B1) by the continuous derivatives: sin φ n+1 − 2 sin φ n + sin φ n+1 = ∂ 2 sin φ ∂Z 2 (B2a) To find the limits of the applicability of this approximation, let us consider continuum approximation in the broad sense and generalize, say, (B2b) to We realize that if shunting is strong, that is either C J /C ≫ 1 or Z J /R J ≫ 1 (the condition implied in this paper), the continuum approximation (in the narrow sense) can be justified when ∆φ ≪ 1, where ∆φ ≡ |φ 1 − φ 2 |.In fact, from (39) follows that in this case the time scale of the solution is proportional to 1/ ∆φ, if γ ≪ 1, and to 1/∆φ, if γ ≫ 1.So the forth order derivative term in (B3) has an additional ∆φ ((∆φ) 2 ) factor with respect to the second order derivative terms, the sixth order derivative term -an additional (∆φ) 2 ((∆φ) 4 ) factor with respect to the second order derivative terms and so on.
In our previous publication 10 we considered also the case of zero shunting.In this case, even if ∆φ ≪ 1, the continuum approximated has to be upgraded to the quasi-continuum approximation (B5) Thus we were able to study the kinks (and the solitons) in the absence of shunting.
. The second problem: