The Korteweg-deVries (KdV) equation with step boundary conditions is considered, with an emphasis on soliton dynamics. When one or more initial solitons are of sufficient size, they can propagate through the step; in this case, the phase shift is calculated via the inverse scattering transform. On the other hand, when the amplitude is too small, they become trapped. In the trapped case, the transmission coefficient of the associated linear Schrödinger equation can become large at a point exponentially close to the continuous spectrum. This point is referred to as a pseudo-embedded eigenvalue. Employing the inverse problem, it is shown that the continuous spectrum associated with a branch cut in the neighborhood of the pseudo-embedded eigenvalue plays the role of discrete spectra, which in turn leads to a trapped soliton in the KdV equation.
Dedicated to the memory of Ludwig Faddeev
The realization that solitary waves are important dynamical entities goes back to seminal observations and experiments of shallow water waves by Russell.1 Motivated by these observations, Korteweg and deVries2 discovered an important equation in shallow water waves. Indeed the Korteweg-deVries (KdV) equation is a universal nonlinear system which arises whenever there is a balance of weak dispersion and quadratic nonlinearity.3
We write KdV’s equation in the following normalized form:
In the context of water waves, t is the time, x is a spatial coordinate, and u is the fluid velocity. In 1965, Zabusky and Kruskal4 found that solitary waves of the KdV equation had special interaction properties; namely, any two of them interacted elastically. They termed these waves solitons. Soon afterward, Gardner, Greene, Kruskal, and Miura (GGKM)5 associated Eq. (1) with two linear equations. One of these equations is the celebrated time-independent Schrödinger equation of quantum mechanics,
where u(x, t) is a real potential, k2 is the wave energy, and t is treated as a parameter.
For rapidly decaying boundary conditions (BCs), GGKM showed how direct and inverse scattering of (2) could be used to linearize and solve the KdV equation. In particular, they showed that solitons were related to time-independent eigenvalues/bound states of Eq. (2) and obtained pure soliton solutions explicitly. The linearization is in terms of a Gel’fand-Levitan-Marchenko (GLM) integral equation which provides the inverse scattering/reconstruction of the solution u(x, t) to Eq. (1). This method of solution is now called the Inverse Scattering Transform (IST) and considerable research using these techniques has ensued and continues today; cf. Refs. 3, 6, and 7. The analytical underpinnings of the direct/inverse scattering problems associated with Eq. (2) for decaying boundary data can be found in Refs. 8–10.
Most research associated with the KdV equation has been posed on spatial domains with either decaying or periodic boundary values. There is, however, an important related problem, which is Eq. (1) subject to step BCs
where c > 0 is constant and u goes to these limits sufficiently fast; we require that
where H(x) is the Heaviside function. We refer to the increasing boundary condition +c2 case as “step up” and the decreasing boundary condition −c2 case as “step down.” Since the KdV equation is Galilean invariant, it suffices to consider u → 0 as x → −∞; i.e., any nonzero boundary condition u → u0 ≠ 0 as x → −∞ can be made zero through the transformation u(x, t) = u0 + ũ(x − 6u0t, t).
The step problem has been studied by a number of authors. With step down boundary values, the basic direct/inverse scattering theory of the Schrödinger equation was developed over 50 years ago by Buslaev and Fomin.11 Their results were later used to discuss the asymptotic behavior of certain solutions to the KdV equation in Ref. 12. Subsequently the problem was studied by a number of authors in Refs. 13–15 and later by authors in Refs. 16 and 17 with the main aim of developing a rigorous understanding of the direct/inverse scattering of this problem.
In terms of the KdV wave dynamics, pure step down data for t > 0 lead to the development of collisionless or dispersive shock waves (DSWs),18 whereas pure step up data for t > 0 lead to the development of a linear ramp from u = 0 to u = c2 with small associated oscillations.19 The asymptotic development of dispersive shock waves due to multi-step down initial data was considered in Refs. 20 and 21.
Unlike previous research on the step BC problem, we focus on the dynamical situation when solitons/sech2 profiles in addition to a step are initially given. To be concrete, we use as initial data a delta function, a box, or a soliton/sech2 profile located well to the left of a step/Heaviside function. The step problem is different from the decaying to zero problem since we do not have “pure solitons,” i.e., along with solitons there is always an additional continuous spectrum (no reflectionless potential).
In the context of Eq. (1), there are two types of pulses associated with localized initial data positioned well to the left of a step. The first case is that of a pulse with a large enough amplitude that allows it to pass all the way thorough the step. The amplitude is related to the discrete spectra/eigenvalues. These are “proper” eigenvalues and correspond to zeros of the inverse of the transmission coefficient, i.e., poles of the transmission coefficient. Proper eigenvalues are associated with standard solitons that, as mentioned above, propagate through the step with only a suitable Galilean shift (velocity increase) and phase shift. From the IST, we derive the phase shift of the soliton as it passes through the rarefaction ramp that evolves from an initial step up or the DSW in the step down case, with or without other solitons. The phase shift formulae are similar to those derived for the decaying problem with a continuous spectrum.22 As an example, we find the phase shift for a single soliton passing through a step. The J-soliton phase shift can be calculated by similar methods (see Appendix C).
However, if the initial localized profile is not large enough, we find that the inverse of the transmission coefficient has no zeros, i.e., no proper eigenvalues. What we do find in this case are spectral values that are exponentially close to continuous spectra. We term such a point a pseudo-embedded eigenvalue. Such pseudo-embedded eigenvalues are associated with soliton-like pulses which propagate as though they were true solitons for a while, but eventually become trapped inside the rarefaction ramp for the step up initial condition, or the DSW in the step down case; hence, they have interesting physical manifestations. In this paper, we discuss the step up case; the step down case is similar.
Below we show that in the trapped case, the “branch cut” term associated with the continuous spectrum in the inverse scattering problem leads to a contribution that plays the role of discrete spectra. Said differently, the pseudo-embedded eigenvalue leads to a dominant contribution from the branch cut associated with the continuous spectrum that has exactly the form as discrete spectra from a proper eigenvalue. This term gives rise to a pulse which travels uniformly like a soliton until it encounters the rarefaction ramp/DSW where it eventually becomes trapped. The pseudo-embedded eigenvalue provides a spectral interpretation of the trapped soliton.
We point out that in recent experiments,23 solitons have been transmitted through rarefaction waves (step up) and dispersive shock waves (step down); moreover, it has been shown that small amplitude solitons can become trapped in the rarefaction ramp/DSW that develops from step initial data.
In this paper, we concentrate on step up BCs; i.e., Eq. (3) with the positive sign. The analogous theory can be developed for step down BCs; see also additional remarks in the conclusion of this article. These studies were motivated by lectures by Hoefer discussing analytical/experimental research summarized in Ref. 23.
II. SCATTERING/INVERSE SCATTERING THEORY AND KdV SOLITONS
The KdV equation (1) is the compatibility condition (Lax pair) for the following two linear equations:
where k is the spectral parameter and γ is a constant; the potential u(x, t) satisfies the BCs (3) with a plus sign.
Eigenfunctions of (5) are defined by the following BCs:
where k, λ are real and
We take the branch cut of λ(k) to be k ∈ [−ic, ic] and the branch cut of k(λ) to be λ ∈ [−c, c]; then, when and when . From the governing integral equations, the eigenfunctions ϕ, ψ can be analytically continued into the upper half plane (UHP) of k, λ, while can be analytically continued into the corresponding lower half plane (LHP). From Eq. (5) and the BCs (7) and (8), we see that the eigenfunctions are related by
for λ, k real and where the asterisk represents complex conjugate. When k = iκ, κ ∈ [−c, c],
B. Scattering data
The two eigenfunctions are linearly independent for k ≠ 0. Hence, we can write as a linear combination of ψ(x, λ, t) and . Thus, we have the relations, formulated on the left, termed the left scattering problem,
where k and λ are real. The scattering data is given by
where W(u, ) = − is the Wronskian. We remark that from the relation λ2 = k2 + c2, the scattering data a, b can be written in terms of either k or λ, i.e., a = a(k, t) or a = a(λ, t). Similar relations hold for , and we can show , for k real and a(−λ, t) = b(λ, t) for real λ such that |λ| ≤ c. We note that k = iκ, |κ| ≤ c, corresponds to real λ, |λ| ≤ c. This forms part of the continuous spectrum and plays an important role below.
For the left scattering problem (13), the usual transmission and reflection coefficients of quantum mechanics are, respectively,
These correspond to a unit wave denoted by e−ikx propagating into the potential u(x) from x = −∞.
In the decaying problem, a soliton solution of the KdV equation is given by
Solitons are associated with zeros of a(k, t), k = iκ, κ > 0, such that a(k, t) = 0. We also call k = iκ an eigenvalue; it is related to a bound state of the Schrödinger equation (5).
The situation is quite different in the step problem. Consider a soliton initially positioned far to the left (x0 ≪ −1) of a localized step centered at x = 0. A soliton with amplitude parameter κ1 > 0 suggests it has a corresponding eigenvalue k = iκ1 with λ1 given by
When the “incoming” soliton has sufficient size, i.e., the corresponding eigenvalue k = k1 = iκ1 satisfies κ1 > c and a(k1) = 0, λ = λ1 = iη1, η1 > 0, then this is an instance of a proper eigenvalue and corresponds to a soliton tunneling through the rarefaction ramp (as an example, see Fig. 1). An analogous equation was obtained via Whitham theory in the weakly dispersive regime and was termed the “transmission condition.”23 As in the decaying problem, the eigenvalue corresponds to a bound state, i.e., the eigenmodes in Eq. (5) which are square integrable. In this case, we find the phase shift of the soliton.
If, however, the initial soliton/sech2(x) profile is not large enough, then the “soliton” becomes trapped.19,23 We term this a trapped soliton since it does look and travel like the sech2 solution in (17) to the left of the step. In fact we find this mode eventually becomes trapped in the rarefaction ramp that evolves from the step (see Fig. 2), never reaching the top of the ramp. This is unlike a normal, or proper, soliton. Below we show that these trapped solitons correspond to pseudo-embedded eigenvalues which are exponentially close to the continuous spectrum.
For an initial soliton/sech2 profile, delta, or box function, we find that the spectral coefficient a(k) takes the form
with 0 < κ0 < c and a2(iκ0) ≠ 0, where x0 is the initial position of a soliton or delta function or box. The calculations leading to Eq. (19) are discussed in Appendix A. Since we take x0 ≪ −1, ϵ is exponentially small. Although a(k) is not found to be exactly zero anywhere in the upper half plane, it does possesses values of k, , that are exponentially close to the imaginary k axis and thus have properties analogous to those of discrete spectra/eigenvalues. The value k0 ≈ iκ0 provides the spectral meaning of a trapped soliton.
In Sec. III, we discuss the inverse scattering problem in the case of proper eigenvalues and proper solitons. In Sec. IV, we show how the form of a(k) given in (19) leads to a spectral contribution that plays the role of a discrete eigenvalue even though, by assumption, there are no proper eigenvalues/discrete spectra in the GLM equation. The scattering data for three prototypical examples (delta function, box function, and soliton) are given in Appendix A.
III. IST VIA GLM AND SOLITON PHASE SHIFTS
The left scattering problem in Eq. (13) can be transformed into a GLM equation from x to ∞. To do this, we assume ψ has the following triangular form:
Substituting the above representation into Eq. (13), using (10), then dividing by a(k), integrating over with ∫dλ exp(iλ(y − x))/(2π) for y > x, and carrying out requisite calculations yields the following GLM equation:
where the kernel is given by
with being simple zeros of a(λ, 0). These proper eigenvalues correspond to kj = iκj, κj > c. The solution of the KdV equation (1) is recovered from
Using the second linear compatible equation (6), we find the evolution of the scattering data to be
where χ(λ) = 8(λ2 − c2)λ − 4c2λ.
When the eigenvalues are proper (κj > c), solitons will move through the rarefaction ramp and the phase shift can be calculated. For J = 1 and as t → ∞, while neglecting the contribution from the continuous spectrum, the following one soliton solution is obtained from Eqs. (21)–(23):
where with being the phase of the 1-soliton when t → ∞. Thus given a soliton at, say, t = 0 with phase x0, means the phase shift of the 1-soliton passing through a step is . A soliton passing through J solitons and a step can be calculated by similar methods (see Appendix C). The phase shift from t = −∞ to t = ∞ is obtained from the GLM equation formulated from −∞ to x, which we discuss next.
The GLM equation from −∞ to x is obtained from the right scattering problem for ψ in terms of , given by
We assume ϕ has the following triangular form:
Substituting the above representation into Eq. (26), taking into account (10), then after dividing by λa(k)/k, and carrying out analogous calculations to those above yields the following GLM equation:
for x > y with kernel
where kj = iκj, κj > c, are simple zeros of a(k, 0); i.e., all the eigenvalues κj, j > 1 are proper. The third term in Eq. (29) is different as compared with the kernel given in Eq. (22). Furthermore this term is associated with the branch cut in the k−plane and −c ≤ λ ≤ c.
With this GLM equation, the solution of KdV equation (1) is given by
The time evolution of the data is given by
The soliton phase shift of proper solitons as t → −∞ can be obtained from the above GLM equation. Assuming no pseudo-embedded eigenvalues and neglecting the contribution from the continuous spectrum, the following one soliton solution is obtained from Eqs. (28)–(30):
where . Hence the total phase shift from −∞ to ∞ across the step is given by , where is given below Eq. (25). It should be noted that this formula contains the step and its associated continuous spectra contributions which are encoded into the normalization constants ; we also note that the above phase shift formulae agree with numerical simulations.19
IV. TRAPPED SOLITONS AND PSEUDO-EMBEDDED EIGENVALUES
In this section, we show how an initial soliton/sech2 profile with amplitude parameter 0 < κ0 < c can be described using the GLM approach. To this end, we consider the GLM equation from −∞ to x given in (28). We assume no proper eigenvalues; so, we only have two terms, both from the continuous spectrum, the first and last terms in (29). The last term arises due to the branch cut.
We assume a localized initial condition: a soliton/sech2 form with corresponding amplitude parameter κ0, or a box function or a delta function, located well to the left of the step centered at x = 0; we call k0 = iκ0 a pseudo-embedded eigenvalue. Additional solitons/sech2 profiles can be added in a similar manner. The first term in the kernel (29) is small in the neighborhood of this pulse, so we only need to consider the branch cut contribution, i.e., the third term.
The dominant contribution to this integral comes from values of κ near κ0 where a(iκ) is nearly zero. We substitute the form of a(k) given in Eq. (19) (see Appendix A for more details) into the branch cut integral in (29), expand around the point k = iκ0, and insert the time dependence of a(k, t) from Eq. (32) into this term, which we call The dominant contribution is given by the integral
and We note that |a(iκ, 0)|2 given in the third term of the kernel (29) is approximated by Δ(κ) in the neighborhood of κ0. Evaluating the above integral, we find
where α = |a1|0|a2|0| cos(φ1 − φ2)| > 0. Remarkably, this has exactly the same form as that from the discrete spectra in the GLM equation given in (29). The corresponding solution is given by
and is valid when x0 ≪ −1 and the soliton position x − x0 is well to the left of the ramp.
Thus, far to the left of the step (x0 ≪ −1), a soliton-like pulse travels with pseudo-eigenvalue κ0. This soliton/sech2 mode travels unimpeded until it comes into contact with the ramp that emanates from the step up initial condition. This soliton/sech2 becomes trapped by the ramp (see Fig. 2). We refer to this as a trapped soliton. To carry out the details of this long time asymptotic analysis of the trapping from the inverse problem is outside the scope of this paper. The weakly dispersive case is discussed in Ref. 19 where analysis and numerical calculations further show how the soliton becomes trapped in the ramp and never makes it to the top of the ramp.
The scattering/inverse scattering theory associated with the time-independent Schrödinger equation and its relationship to soliton solutions of the KdV equation for step potentials was analyzed.
The first case we considered was that of eigenvalues a(k) = 0, k = iκ, κ > c, where we find “proper” solitons. In this case, the inverse scattering theory and linearization of the KdV equation can be carried out via a GLM equation with the solitons calculated from the discrete spectrum of the GLM kernel. Here a soliton that is initially well separated from the step propagates all the way through a ramp; doing so, it acquires a phase shift which can be calculated exactly. This phase shift has encoded in it the continuous spectra which arises from the step. Numerical calculations confirm these formulae.19
The second case was that of spectral data which had no proper eigenvalues, yet behaved as though it did. In terms of the soliton pulse, the amplitude is not large enough to pass through the rarefaction ramp that develops from the step up initial condition. This becomes a trapped soliton. In spectral terms, there is a point, k = iκ, 0 ≤ κ ≤ c, where the inverse of the transmission coefficient, a(k), is exponentially close to, but not, zero. In this case, the continuous spectrum associated with the branch cut gives rise to a contribution that approximates a discrete eigenvalue located at k = iκ0. We call such κ0 a pseudo-embedded eigenvalue.
Although the analysis here is developed for step up boundary conditions, the step down case is similar. In Ref. 23, the correspondence between the step up and step down case is referred to as “hydrodynamic reciprocity.” In the step down case, a localized initial profile is inserted to the right of the step. Upon evolution, it gets trapped by a DSW that emanates out of the initial data. From a mathematical viewpoint, we have the relationship λ2 = k2 − c2 for step down boundary data [compare this with Eq. (9)]. Here the scattering/inverse scattering theory corresponds to interchanging the roles of λ and k.
We are pleased to contribute this article to the special volume in honor of L. Faddeev. Faddeev made many far reaching/major contributions to mathematical physics. One of his earliest interests involved the study of scattering theory of the one dimensional Schrödinger equation with a decaying potential. The present article, motivated by recent experiments, involves the study of the Korteweg-deVries equation and its linearization via the inverse scattering transform with this Schrödinger equation, now with step boundary values. Although the direct and inverse scattering theory has its roots going back many years, it is still a source of keen interest.
M.J.A. was partially supported by NSF under Grant No. DMS-1712793.
APPENDIX A: EXAMPLES OF SCATTERING DATA
In this appendix, we discuss three examples of potentials associated with the time-independent Schrödinger equation (2). These potentials are given as initial conditions associated with the KdV equation (1) at t = 0. Each consists of a localized hump well separated from a step with the form
where the Heaviside function H(x) is given by
Here x0 represents the center of the localized hump and is assumed to be located far to the left of x = 0. Below we denote these initial functions as .
1. Soliton/sech2 potential
Consider a soliton/sech2 profile positioned well to the left of the step function
where −x0 ≫ 1, c > 0, κ0 > 0.
If we consider the decaying problem (c = 0) corresponding to the above sech2 potential, we can calculate ϕ(x, k), ψ(x, k) exactly. The reason for this is that when b(k) = 0 in Eq. (13), or when in Eq. (26), there are significant simplifications. In Eqs. (13) and (26), we divide by a(k), subtract the pole contributions, use the symmetries in (10), and take a minus projector. Evaluation at k = iκ0 yields the bound state and then using the bound state for general k we can calculate the eigenfunction (cf. Ref. 3). In this way, we find the eigenfunction ϕ(x, k) which is valid for x ≤ 0 in the step problem; similarly, we can get ψ(x, k). However for ψ(x, k) in the step problem, k is replaced by λ since the spectral parameter satisfies λ2 = k2 + c2. The results are
From these results, we can calculate the scattering data using the Wronskian. Using Eq. (15), a(k) is found to be
From the above relation, we can calculate the first two terms of the asymptotic approximation of the form given by Eq. (19), which for convenience we give again
The values a1(k), a2(k) for the soliton plus step (A8) are given by
To approximate the branch cut integral in (29) in the case of pseudo-embedded eigenvalues, we focus on values of k near iκ0 since that is where a(k) is at a minimum. As such, to get the asymptotic integral in Eq. (34), we expand a(k) around k = iκ0 which results in evaluating a1(k) and a2(k) in Eq. (19) at k = iκ0.
We remark that a similar example can also be calculated exactly, namely, that of a soliton truncated at zero at x = 0,
In this case, ϕ(x, k) is still given by Eq. (A4) and ψ(x, λ) = eiλx for x ≥ 0. Hence a(k) can be calculated from the Wronskian relation. The formula is
which has the same a1(k) as (A7), but different a2(k).
We also note that while there are solutions a(k) = 0 for , we do not find any solutions to a(k) = 0 when . An asymptotic expansion suggests that the zeros of a(k) are complex, i.e., k0 = ξ0 + iκ0 where κ0 > 0, ξ0 ≠ 0. This is, in fact, a contradiction since any eigenvalue corresponding to a bound state must be purely imaginary (see Appendix B).
2. Delta potential
Consider a delta function of height Q positioned well to the left (−x0 ≫ 1) of a step function,
The time-independent Schrödinger equation can be explicitly calculated in this case. The solution takes the form
for At x = x0, the eigenfunction ϕ(x, k) satisfies the jump condition and continuity. Using continuity of ϕ(x, k) and its derivative at x = 0 yields the remaining coefficients. We only give a(k) for the delta function plus step below
In the case of proper eigenvalues, there exists k1 = iκ1, κ1 > c such that a(k1) = 0. For −x0 ≫ 1, κ1 ≈ Q/2, which is the same as the decaying (non-step) problem. When 0 < κ0 < c, the unperturbed problem, i.e., without the exponential term in Eq. (A12), suggests that κ0 should be approximated by Q/2. But keeping the exponential term and carrying out an asymptotic expansion for −x0 ≫ 1 leads to the zeros of a(k) being complex; i.e., k0 = ξ0 + iκ0 where κ0 > 0 and ξ0 ≠ 0. This is a contradiction since any true bound state eigenvalue is purely imaginary (see Appendix B). In fact trying to solve a(k) = 0 numerically does not lead to a convergent iteration for this or any of the pseudo-embedded/trapped soliton examples discussed in this appendix.
3. Box potential
Consider a box function positioned well to the left (x0 ≪ −1) of a Heaviside function (A2),
with height h2, h > 0, and width L > 0. The solution takes the form
where . We enforce continuity of the solution and its derivative at x = x0 and x = 0. All coefficients can be calculated. We only give a(k) for the following case:
If we neglect the term multiplied by , i.e., take x0 ≪ −1, the eigenvalues satisfying a(k) = 0 yield solutions for satisfying
Solutions to the above equation are the same as those obtained in the decaying no-step problem.3 Graphical analysis shows that there can be one or more solutions depending on the size of h and L. For example, when h < π/L, there is one solution, which we denote as κ0 for 0 < κ0 < h.
Keeping the exponential term modifies the above result. For , define . Solutions of a(k) = 0 must satisfy
Note that both left- and right-hand sides of the above formula have unit modulus and hence a perturbative solution for −x0 ≫ 1 is expected; numerical solutions have been found.
For , solutions of a(k) = 0 now satisfy
In this case, the left-hand side is not of unit magnitude; no solution is expected; a numerical solution has not been found. The values of a1(k) and a2(k) can be found from Eq. (A16).
APPENDIX B: BOUND STATES
In this appendix, we establish that all solutions of a(k) = 0 associated with bound states, i.e., eigenvalues, must be purely imaginary with . Recall that
and for the Wronskian W(ϕ, ψ) = ϕψx − ϕxψ. If a(k0) = 0 and k0 = ξ0 + iκ0 for ξ0 ≠ 0, then ϕ(x, k0) = β0ψ(x, k0) for some nonzero constant β0. Thus,
where . Assuming that
then necessarily and hence κ0 > c. Such eigenvalues are proper. Moreover, we point out that ϕ(x, k) and its complex conjugate ϕ*(x, k) satisfy the equations
respectively, with u(x) real. Hence,
Since ϕ → 0, ϕx → 0 as x → ±∞, we have
If a(k0) = 0, where k0 = ξ0 + iκ0, then
for , so and ξ0κ0 = 0. For decay as |x| → ∞, we require κ0 > 0; thus ξ0 = 0.
APPENDIX C: PHASE SHIFT FOR J SOLITONS
1. J-soliton solution from the GLM equation (21)
We take J ordered proper eigenvalues, i.e., J distinct simple zeros of a(λ). We assume that each soliton is initially well separated and located well to the left of the step. Neglecting all effects due to the continuous spectrum, the kernel Ω in the GLM equation (21) is
defines the phase of Jth soliton when t → +∞.
2. J-soliton solution from the GLM equation (28)
As above, we take J eigenvalues, i.e., J distinct simple zeros of a(k), a(kj = iκj) = 0. We assume that each soliton is initially well separated from every other soliton and neglect the effects due to the continuous spectrum, the first and third terms in the kernel Ω are given in Eq. (29). We consider J ordered solitons that are initially separated well to the left of the step. The kernel is given by
where κj > 0 and κ1 < κ2 < … < κJ where is defined below (29). As , the Jth soliton satisfies
defines the phase of Jth soliton when t → −∞. Thus, the total phase shift of Jth soliton due to the step and other solitons is given by