Exact Solutions of Semiclassical Non-characteristic Cauchy Problems for the Sine-Gordon Equation
The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter tends to zero. Assuming natural initial data having the profile of a moving kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small . This sequence of exact solutions is analogous to that of the well-known -soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. Plots of exact solutions for small reveal certain features that emerge in the semiclassical limit. For example, in the limit one observes the appearance of nonlinear caustics, i.e. curves in space-time that are independent of but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases.
In the appendices we give a self contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic -Sobolev initial data, and Appendix B establishes the well-posedness for -Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).
The sine-Gordon equation
describes a broad array of physical and mathematical phenomena. The partial differential equation (1) may be regarded as the continuum limit of a chain of pendula subject to an external (gravity) force and coupled to their nearest neighbors via Hooke’s law. In nonlinear optics, the sine-Gordon equation is a special case of the Maxwell-Bloch equations and describes self-induced transparency in the sharp-line limit . In biology, the sine-Gordon equation models transcription and denaturation in DNA molecules . Bäcklund showed a correspondence between solutions of the sine-Gordon equation and surfaces of constant negative curvature .
In solid-state physics, the sine-Gordon equation models idealized magnetic flux propagation along the insulating barrier between two superconductors in a Josephson junction. Here the length of the transmission line corresponds to a length of in terms of the dimensionless coordinate measuring distance along the junction. Let be the inductance per unit length and be the capacitance per unit length. Then is the typical velocity parameter, and the macroscopic time scale measures one dimensionless unit when seconds have passed. The parameter is the ratio of the Josephson length to the transmission line length . The Josephson length is in turn proportional to , where V sec is the quantum unit of magnetic flux. Laboratory experiments by Scott, Chu, and Reible  analyzed flux propagation in Josephson junctions of length cm for which was approximately to m. Therefore, in these experiments, . The period of a signal input to the transmission line in these experiments was typically on the order of seconds, which is approximately one dimensionless time unit on the -scale. Together with being small, this motivates the study of the semiclassical (or zero-dispersion) limit as . For analytical convenience we choose to study the Cauchy initial-value problem on the real line . Formulating a semiclassical Cauchy problem means fixing functions and independent of , and then, for all sufficiently small, posing the Cauchy problem for (1) with initial data of the form , . See Appendix B for an account of the well-posedness theory of this Cauchy problem for fixed. Solving the semiclassical Cauchy problem means obtaining the one-parameter family of solutions . We are usually most interested in the asymptotic behavior of the solution as .
In this paper, we consider the sine-Gordon equation (1) for all sufficiently small with the initial condition
where is a parameter. We refer to the solution of the Cauchy problem as . The topological charge (or winding number) of solutions satisfying (2) is a constant of motion given by
From one point of view, the initial data (2) are natural to study, because is an exact mathematical antikink solution of the sine-Gordon equation explicitly given by
with velocity and width parameter given by
From these formulae we see that is a traveling wave with velocity bounded by 1 (the light speed), demonstrating the hyperbolicity of the sine-Gordon equation. This solution admits a natural relativistic interpretation since the relationship between and corresponds to Lorentz contraction in special relativity.
For , the initial data (2) no longer corresponds to simply one soliton, but in general excites a nonlinear superposition of kinks, antikinks, breathers, and radiation. It is interesting to observe that the initial data (2) satisfies the advection equation with constant velocity . In this sense, we may consider the initial data as being in uniform motion to the right with velocity . Note, however, that if , then for sufficiently small, the velocity of the initial data exceeds the constraint imposed by the hyperbolic nature of the sine-Gordon equation (1). In this situation, one might expect the sine-Gordon equation to regularize the superluminal velocity of the initial data for by some kind of catastrophic effect that destroys the profile of the initial data. In fact, we will show (see figures 6 and 8) that the regularization of the velocity takes place via the emission of a large number (inversely proportional to ) of kink-antikink pairs.
The family of solutions corresponding to the initial data (2) may be viewed as an analogue for the sine-Gordon equation of the -soliton (or higher-order soliton) solution to the cubic focusing nonlinear Schrödinger (NLS) equation
Satsuma and Yajima  found that with initial data = the scattering data relevant for the focusing NLS equation can be found in closed form for any . Furthermore, if then the scattering data are reflectionless and so the solution can be found more-or-less explicitly. In  it was noted that, with and , the function satisfies the initial-value problem
The functions therefore solve a semiclassical Cauchy problem since is independent of . Numerical reconstruction of the inverse-scattering solution for , in  revealed a spatio-temporal pattern for emerging as consisting of a fixed macrostructure with nonlinear caustics (phase transition boundaries or breaking curves) separating regions of the space-time plane consisting of oscillations of different local genus (number of nonlinear phases). At least two caustic curves appear in the dynamics (a primary caustic and a secondary caustic ). The semiclassical asymptotics for times up to and just beyond the primary caustic were obtained in  and these results were extended to times just beyond the secondary caustic (requiring a substantial modification of the method that captures the primary caustic) in . In a related result, Tovbis and Venakides  generalized the calculation of Satsuma and Yajima by computing the scattering data associated with the semiclassically-scaled focusing NLS equation (7) explicitly for all sufficiently small when the initial data is given in the form
Subsequently, the Cauchy problem for (7) with this initial data has been studied by Tovbis, Venakides, and Zhou [22, 23]. In this paper, we present a calculation of the scattering data for (1)–(2) analogous to the work in  and , and we also present an explicit computation of as analogous to . The asymptotic analysis of the semiclassical Cauchy problem for sine-Gordon corresponding to the work in [12, 15, 22, 23] will be carried out in a later work.
The sine-Gordon equation (1) is an integrable system, possessing a Lax pair (see (123) and (124)) and admitting all the benefits thereof, including the existence of inverse-scattering transforms for solving Cauchy problems in various coordinate systems. We consider the Cauchy problem in laboratory coordinates and we use the Riemann-Hilbert formulation of inverse scattering. For the sine-Gordon equation in characteristic coordinates, the inverse-scattering method was first given in  and . The inverse-scattering method corresponding to the (noncharacteristic) Cauchy problem for the sine-Gordon equation in laboratory coordinates was worked out by Kaup . An account of the Riemann-Hilbert method for carrying out the inverse step in laboratory coordinates can be found in the text of Faddeev and Takhtajan , and further developments to the theory were made by Zhou  and Cheng et al. . In our paper, we add to this literature by giving in Appendix A a complete description of the Riemann-Hilbert formulation of the solution of the Cauchy problem in laboratory coordinates assuming that at each instant of time the solution has -Sobolev regularity. That the sine-Gordon equation (1) preserves this degree of regularity if it is present at is established by independent arguments in Appendix B.
Briefly, the inverse-scattering method proceeds as follows. Cauchy data for the sine-Gordon equation characterize a set of scattering data, which consist of the reflection coefficient , the eigenvalues , and the modified proportionality constants . The scattering data are used to formulate a Riemann-Hilbert problem with an explicit, elementary dependence on and . While it is not in general possible to solve a Riemann-Hilbert problem in closed form, for reflectionless Cauchy data (i.e. for which ) the Riemann-Hilbert problem can be reduced to the solution of a system of linear algebraic equations.
In Section 2, we explicitly calculate the scattering data corresponding to viewing the initial data (2) as a kind of potential in the linear scattering problem (123) associated with the Cauchy problem for the sine-Gordon equation (1). Our analysis will be valid for all and sufficiently small. Furthermore, we show that if lies in the sequence
(note that this sequence converges to zero as ), then the scattering data are reflectionless ultimately implying via inverse-scattering theory that can be computed explicitly (that is, can be expressed by a finite number of arithmetic operations). The inverse step is carried out for in the sequence (9) corresponding to reflectionless initial data in Section 3, where and are extracted by considering an appropriate limit of the solution to the Riemann-Hilbert problem. As through this sequence, a pattern emerges in which consists of modulated wave trains of wave number and frequency inversely proportional to with one or more nonlinear phases. The spatio-temporal scale of the modulation is fixed as . Regions of space-time containing waves with different numbers of nonlinear phases are separated by nonlinear caustics that are independent of for fixed . See figures 5, 6, and 8 for plots of exhibiting these features for various values of and . At a qualitative level, these features resemble those observed for solutions of the semiclassical Cauchy problem for the focusing NLS equation. Section 4 discusses the limitations inherent in an approach to the semiclassical limit based upon calculations of complexity and sensitivity increasing with , and explores possible extensions.
where . This is a fixed-dispersion Cauchy problem with a sequence of different initial conditions depending on , just as in the problem for the NLS equation studied by Satsuma and Yajima. The initial conditions all have topological charge but and become more dilated in (slowly-varying) as increases. Therefore, an alternate way of viewing our result is that we can find exact solutions to the fixed-dispersion initial-value problem (10)–(11) for . As an example of an explicit solution of (10) obtained in this way, when and we have
The focusing NLS equation (6) admits a scaling symmetry in which scaling the independent variable is equivalent to scaling the dependent variable (amplitude) and the time . Thus, the -soliton (or higher-order soliton) solutions of the focusing NLS equation that were originally obtained by Satsuma and Yajima  by considering a fixed-width pulse with variable amplitude can just as easily be viewed as a fixed-amplitude pulse with variable width. From the point of view of semiclassical asymptotics, dilation in is the more natural interpretation of the higher-order solitons as the presence of the parameter in (7) amounts to rescaling and , and thus the variable width of the pulse is absorbed into the semiclassical parameter as in [17, 12]. Of course, the sine-Gordon equation does not admit the amplitude/dilation symmetry enjoyed by the focusing NLS equation, so we are not free to interpret the family of exact solutions we obtain in this paper in terms of scaling of amplitude. It seems that perhaps a more generally fruitful approach to seeking analogues of the higher-order soliton in other integrable systems is to consider pulse width dilation as being more fundamental than amplitude dilation. As more evidence of the utility of this approach (beyond the sine-Gordon example), the modified NLS equation (which includes an additional term in (6) that breaks the scaling symmetry needed to exchange amplitude for width) does not have higher-order solitons in the sense of Satsuma and Yajima, but it does have exact solutions corresponding to arbitrarily width-dilated pulses that are useful in semiclassical analysis .
Remark 2. In characteristic or light-cone coordinates
and defined by and , the
sine-Gordon equation (1) is
, where . The
associated evolution equation in the Lax pair is the
Zakharov-Shabat eigenvalue equation, which is the same eigenvalue
equation as for the focusing NLS equation . Thus
it is possible to solve a semiclassical characteristic Cauchy problem
with special initial data using the
Satsuma-Yajima higher-order soliton solution. However, in many
applications (as in Josephson junction theory), the correct problem is
the non-characteristic Cauchy problem with two independent initial
conditions: , . The
Satsuma-Yajima solution to the semiclassical problem posed along a
characteristic or will have a very complicated and
unwieldy form and an undesired dependence on upon restriction to
, and therefore is probably not relevant to the
Cauchy problem we wish to consider.
On notation. As will be explained in detail in Section 2 and Appendix A, we will use three different gauges for the eigenvalue problem. Objects associated with the infinity gauge will be denoted by an overline (). Objects associated with the zero gauge will be denoted by an underline (). Finally, objects associated with the symmetric gauge will not have a bar (). The complex conjugate of is denoted by . We use the notation to emphasize that is a parameter. The dependence on parameters may be suppressed by writing in place of . We also make frequent use of the standard Pauli matrices defined as
Vectors will be denoted by bold lower-case letters and matrices by bold upper-case letters, with the exception of the Pauli matrices. The transpose of a vector is denoted by , and the conjugate-transpose of a matrix is denoted by . Finally, denotes the characteristic function (indicator function) of a set , that is if and otherwise.
2. Scattering Theory for the Special Initial Data
The quantities appear throughout the scattering and inverse-scattering theory of the sine-Gordon equation (1), and so for convenience we define
This formulation of the eigenvalue problem is useful in the study of solutions when is bounded away from (see  and Proposition A.1 below), and for this reason, we say that (16) is written in the infinity gauge. The use of alternate gauges proves to be beneficial. For example, the gauge transformation (139) (see Appendix A) casts the eigenvalue problem (16) into an alternate form that is useful in the analysis of solutions corresponding to bounded , and in particular near (see  and Proposition A.2). Therefore, we refer to the coordinate system arrived at via the transformation (139) as the zero gauge. While the infinity gauge and the zero gauge are useful in the analysis of the scattering problem required to formulate an inverse-scattering theory, to calculate the scattering data corresponding to (2) we found it to be useful to introduce a gauge transformation that symmetrizes the appearance of and in the eigenvalue problem and at the same time also removes the function from the coefficients. It is in this third, symmetric gauge that it is easiest to see the eigenvalue problem is in fact hypergeometric for the initial data (2). Once it is clear from working in the symmetric gauge that the eigenvalue problem has exactly three regular singular points, it is possible to use the theory of Euler transforms to analyze the asymptotic behavior of the Jost solutions and thus obtain the scattering data.
2.1. Transformation to a hypergeometric equation
The first step in transforming (16) into a hypergeometric equation is to introduce an appropriate gauge transformation. If satisfies equation (16), then the invertible transformation (having an interpretation as a rotation at each by an angle )
yields a solution of the eigenvalue problem written in the symmetric gauge:
Written in this form111The absence of in the symmetrized form (18) of the eigenvalue problem provides an alternate framework in which to consider discontinuous initial data without the use of delta functions (cf. )., the eigenvalue problem appears similar to one used by Faddeev and Takhtajan (see  part 2, chapter 2, equation 4.1). The Jost solutions in the symmetric gauge are defined to be the fundamental solution matrices of the linear problem (18) for real values of , normalized by the conditions
We denote the columns in this way: . They are related to the Jost solutions for the infinity gauge (see (125)) by
With the change of independent variable
the eigenvalue problem (21) becomes
Here and the positive square root is chosen. There are two ways to eliminate the square roots in the coefficient matrix. The first is to introduce the linear transformation
which results in a differential equation satisfied by :
This equation has exactly three regular singular points and can be written in hypergeometric form. We will use (25) to find expressions for and .
Remark. If we had taken (16) instead of (18) as our starting point and followed analogous steps, namely (i) substitution of the initial data using double-angle formulae, (ii) the independent variable transformation , and (iii) the use of the gauge transformation (24) to reduce the problem to rational form, we would have arrived at
instead of (25). Let . Then near , (26) has the leading-order form , whereas (25) has the leading-order form . After some calculation it is possible to see that the method of Frobenius still applies to (26) near even with the additional growth at due to special identities holding among the entries of the matrix coefficients of the leading-order terms. However, the local (Frobenius) analysis of (25) is more straightforward with only a double pole at . Later we will also see that it is more difficult to obtain integral representations for solutions of (26) than for (25).
2.2. Integral representations for Jost solutions
We now use the theory of Euler transforms  to derive integral representations for the four Jost solutions, starting with and . Define
Choose the principal branches of the functions with branch cuts on the real -axis from to . Also choose the principal branch of with branch cut on the real -axis from to . Take to be a closed contour in the -plane passing through the branch point and encircling once in the counterclockwise direction
(see figure 1(a)). Then, for , is given by
We begin by computing the Frobenius exponents of (25). Assume that, for some number , has a Frobenius series about of the form
for some vector-valued coefficients . Substituting this series into equation (25) and considering the leading-order terms immediately gives the (indicial) eigenvalue equation
Therefore, the Frobenius exponents at are
Similarly, substituting a series of the form
into equation (25) and considering the leading-order terms shows that the Frobenius exponents at are exactly the same:
We now shift two of the exponents to zero via the substitution
It follows that satisfies the differential equation
We attempt to express as
Remark. The fact that there is such a simple relationship between and is related to the fact that could be easily eliminated to write a second-order differential equation for that is essentially the Gauss hypergeometric equation. On the other hand, if we had worked from the beginning in the infinity gauge, the elimination of using the first row of (26) would have resulted in a second-order equation that is not obviously of hypergeometric form. This, in turn, would lead to further complications in the following analysis leading from (2.2) to (49).
It remains to satisfy the second equation of the system (39):
Writing and and using equations (41) gives
If we now choose to satisfy the quadratic equation
then the terms will cancel in equation (2.2). Specifically, we choose
Using integration by parts to eliminate the term yields
Setting the integrands equal and using equation (45) for gives the first-order linear differential equation
for . The general solution
for some constant , as is bounded away from and . Since , the function is integrable on . Therefore, by Lebesgue’s dominated convergence theorem,
The integrand is integrable at , so we can collapse to the upper and lower edges of the branch cut , yielding
In the last step we used the reflection identity
The remaining integral is a beta integral, which may be expressed in terms of gamma functions. Indeed, making the change of variables gives
using the identity
valid for . Also note that, as ,
Therefore, as ,
Choose the principal branches of the functions with branch cuts on the real -axis from to . Also choose the principal branch of with branch cut on the real -axis from to . Take to be a closed contour in the -plane passing through the branch point and encircling once in the counterclockwise direction (see figure 1(b)). Then, for , is given by
with given by equation (32).
The construction follows that of , except with in place of , and choice of the solution
to equation (48). ∎
Take , , and as in Proposition 2.1. Then, for , is given by
The Frobenius exponents around and around for satisfying (28) are
Therefore, defining in terms of by
has the effect of shifting one exponent to zero near each of the points . By direct calculation, satisfies
Assume integral representations of the form
Proceeding as in Proposition 2.1, we obtain