1. Introduction

The study of solutions of the Korteweg–de Vries equation

with initial conditions having different asymptotics for x → ±∞ was initiated by Gurevich and Pitaevskii [12, 13] who obtained a description of the asymptotic behaviour by use of the Whitham averaging method. The approach based on the inverse scattering transform was developed by Hruslov and Kotlyarov [14, 17], and the well-posedness of the Cauchy problem was justified by Cohen [6] and Kappeler [16], under assumption that step-like initial data tend to the limiting values fast enough. Further intensive study of this problem led to many important results, including the refinement of asymptotic formulae, generalizations for the finite-gap type of asymptotic conditions, and generalizations for other nonlinear equations, see e.g. [3, 8, 21, 28] and references therein.

Despite these successes, no explicit step-like solutions have yet been found (to the author’s knowledge), similar to solitons in the rapidly decaying case. In fact, it is hardly possible that such a solution can be given by elementary functions or even classical special functions. The goal of the present paper is to show that an answer can be given by some ordinary differential equation of Painlevé type. It should be noted that a similar result was previously obtained by Suleimanov [25] for another Gurevich–Pitaevskii problem, also posed in [12, 13], on the formation of the oscillating zone in the vicinity of the breaking point. The exact answer proposed in [25] was defined as the stationary equation for the linear combination of the 5-th order KdV symmetry and the classical Galilean symmetry, which yields the following ODE consistent with (1.1):

This equation is also one of the representatives of the so-called string equations [18,20]. In general, its real solutions have pole singularities, but there is also a special solution which is regular for all real x and t and has the asymptotic behaviour for |x| → ∞ defined by the equation without derivatives, that is, by the cubic parabola 10u³ + k(6tu + x) = 0. The conjecture on the uniqueness of this solution was formulated in [7] and a rigorous proof of the existence was obtained in [5]. The properties of equation (1.2) and its applications to the Gurevich–Pitaevskii problem were studied in [10, 19, 26, 27].

In this paper we demonstrate, so far only numerically, that the Cauchy problem with the step-like boundary conditions also admits some exact solutions related to nonautonomous symmetries. To this end, we consider the stationary equation for a linear combination of the master-symmetry and the scaling symmetry which is equivalent to some sixth order ODE. Under suitable choice of parameters, this equation admits two different constant solutions which suggests that a regular solution may exist with different asymptotic for x → ±∞. As in the case of the equation (1.2), the difficulty is that the general solutions of this ODE may have singularities, while the regular solutions form a low-dimensional submanifold; so, even their existence is far from obvious.

The final result of our experiments is illustrated by Fig. 1, where we compare our solution with the solution of the Cauchy problem with a kink-like initial condition (which is viewed as an approximation of the original Gurevich–Pitaevskii setting with the unit step function). One can see a good agreement for the compression waves, but there are also some peculiar features of our solution which should be noted:

Fig. 1.

Solution of eqs. (3.3), (3.4) vs. solution of the Cauchy problem with initial condition u(x,0)=12(tanhx−1).

• — the initial condition is not monotonic;

• — the approach to limiting values is slow (as O(x−²) for u → 0 and O(x−¹) for u → 1);

• — there are no oscillations in the lower part of the rarefaction wave; instead, there is a slow decay of the initial oscillations in the upper part.

The paper is organized as follows. In section 2 we recall several basic facts about the symmetry algebra of the KdV equation and introduce our sixth order stationary equation (2.1). In section 3 the zero curvature representation is given, based on the well known interpretation of master-symmetry as the flow changing the spectral parameter, see e.g. [2, 4]. This representation provides two first integrals, as well as a more convenient form of equations (3.3), (3.4) which are the main object of further study. In section 4 we discuss, very briefly, a typical scenario of blow-up for the generic solutions of these equations. It should be noted that the efforts in study of Cauchy problems are mainly aimed at the search of the well-posedness conditions, see e.g. [24] where several non-standard settings were discussed. However, the singular solutions are also very interesting, although less studied. Some sufficient conditions for the blow-up of KdV solutions were obtained in [23].

Section 5 contains a description of regular solutions satisfying the fourth order ODE at the special line t = 0 which is the fixed singular line for equations (3.3), (3.4). Taking two first integrals into account, one can reduce this equation to a special case of P₅ equation. In turn, this equation has the fixed singular point x = 0 and this effectively lowers the order once again, so that the dimension of regular solutions submanifold is 3. In section 6, we consider the limiting transition to the separatrix solutions at the boundary of this submanifold which finally gives us the desired step-like solutions.

2. Stationary equations for nonautonomous symmetries of KdV

It is well known that the algebra of higher symmetries of the KdV equation is obtained by use of the recursion operator R=Dx2+4u+2u1Dx−1 :

All flows of order ⩽ 5 read

where the non-local variable u₋₁ in the latter equation is differentiated according to the rules

Equation for u_τ5 was introduced in [15], see also [4,9,22] for the general notion of master-symmetry. Both sequences ∂_ti and ∂_τi can be continued infinitely, which gives us the commutative and non-commutative parts of the hierarchy, respectively. However, in this paper we do not need an explicit form of the higher flows. The stationary equation E[u] = 0 for any linear combination of symmetries defines a constraint which is compatible with (1.1) due to the identity

It is well known that the use of only autonomous symmetries leads to the algebro-geometric solutions, and adding any nonautonomous one brings to the Painlevé type equations. For instance, the Suleimanov equation (1.2) is equivalent to u_t5 + ku_τ1 = 0. Let us take a general linear combination of the flows written above, containing the master-symmetry:

The group of classical symmetries acting on this equation makes possible to fix all the coefficients, which leads to three nonequivalent cases. First of all, we set k₃ = k₄ = k₅ = 0 without loss of generality, by adding constants to t,x and u₋₁. This gives the equation

which is the sixth order ODE with respect to u₋₁, with t playing the role of parameter (in the next section, we will rewrite this equation in a bit more convenient form). Next, the coefficients k₁, k₂ can be changed by use of the following transformations.

3. Zero curvature representation and first integrals

Let us recall the scheme for obtaining the KdV symmetries from linear matrix equations of the form

where K = K(λ) is a polynomial with constant coefficients, see e.g. [2,4]. The KdV equations itself is equivalent to the compatibility condition of the first pair of equations under the choice

The symmetries are obtained from the compatibility conditions with the third equation:

One can easily prove that the general form of the matrix W is

(this is also true for U and V, with Y = 1=2 and Y = −2λ +u, respectively) and that the compatibility conditions amount to equations

Assuming that Y is a polynomial in λ of degree no lower than that of K, we can determine all its coefficients step by step. The stationary symmetry equation has the representation

which is equivalent to

In the case K = 0 corresponding to autonomous symmetries, equations (3.2) have the common first integral polynomial in λ:

If degY = n then the coefficients of this polynomial give us 2n first integrals which turns out to be enough for the Liouville integrability.

In the case K ≠ 0, only deg K first integrals survive (which is too small to speak about the Liouville integrability), corresponding to the zeroes of K, counting with multiplicities:

In relation to our equation (2.1) with k₁ = −4 and k₂ = 0, that is, the stationary equation u_τ5 −4u_τ3 = 0, we have

It is convenient to rewrite the equations in terms of the variables u and y.

4. Blow-up of generic solutions

In equation (3.3), the coefficient at the higher derivative u₃ is equal to t. Therefore, t = 0 is the fixed singular line for the pair (3.3), (3.4); a generic common solution of these systems acquires a singularity along this line. The explicit solution u = −x/(6t) gives a simplest example of the pole, but, in general, the structure of singularity is much more complicated.

In the half plane t < 0 (or t > 0), the generic solution can be constructed in two ways, given initial data (u₀, u₁, u₂, y₀, y₁, y₂) ∈ ℝ⁶ at a point (x₀,t₀), t₀ < 0. One way is to solve first the system (3.4) for fixed x = x₀. This gives us the initial data for the system (3.3), which we then have to solve for all t ∈ (−∞,0). Another way is to solve first the system (3.3) for fixed t = t₀, and then to use the solution as initial data for (3.4), for all x. Both construction methods give the same result due to the commutativity of the flows.

Of course, the constructed solution may have movable poles for t ≠ 0 as well. However, numerical experiments show that for some region in the space of initial data the solutions do not have singularities for x ∈ ℝ and t ∈ ℝ_<0 (or t ∈ ℝ_>0). For x small enough, a typical solution of such kind looks like small oscillations near u = 1, but, in fact, it grows in x on one half-line (see Fig. 2 where two plots on the top correspond to the same moment t, but have different scales in x). When t increases, the growth in x becomes more noticeable, and at the same time, the amplitude and frequency of oscillations increase on the other half-line. For t → 0 this leads to a pole-like singularity on one half-line and an essential singularity on another, as shown on Fig. 3.

Fig. 2.

Formation of a singularity at t → 0 for a generic solution.

Fig. 3.

A portrait of generic solution. The initial data are given at x = 0, t = − 5; thick lines show the solution of (3.3) for t = −5 and the solution of (3.4) for x = 0.

5. Equation for t = 0 and regular solutions

Further on we consider only special solutions which remain regular for t = 0. For these solutions the order of equation (2.1) on this line lowers and it turns into the fourth order ODE (we assume that k₁ = −4 and k₂ = 0)

It should be emphasized that the decrease in order occurs only along the special line t = 0; outside it, any regular solution is described by the sixth order equation, as before. The equivalent system (3.3) and its first integrals take the following form, in the prime notation for the derivatives:

This yields the second order equation for y after elimination of u:

It can be brought to the special case of Painlevé equation P₅ with vanishing parameter δ [2] which, in turn, is equivalent to a special case of P₃ [11].

6. Limiting transition to step-like solutions

By smoothly varying the initial data for the system (5.1), one can notice a broadening of the well near the origin at the moment preceding the formation of a pole. During this elusive moment, the solution remains practically unchanged on one half-axis, but changes drastically on the another one, so that the oscillating zone moves farther and farther from the origin. How far can it be driven away?

Let us use the shooting method to select the initial value y(0) = a₀, either for fixed a₁,a₂ or for fixed values of the first integrals (5.3). Let an interval [a₀(1),a₀(2)] be given such that the solution corresponding to one endpoint has a pole, and the solution corresponding to another is oscillating. Then we compute the solution at the midpoint and select half the interval so that its endpoints correspond to solutions of different types, as before. Repeating this yields a sequence of values

for which the well is gradually widening. In the limit, a separatrix solution in the form of a step arises, as shown in Fig. 5 (of course, not all intermediate solutions are shown, there are about 100 ones in this example). This is a process reminiscent of the limiting transition from a cnoidal wave to a soliton, but it affects the distance only between two peaks.

Fig. 5.

Streching of the step, at H₀ = −2, H₁ = −6.

Roughly speaking, to expand the well by 1, we need to calculate the next exact decimal digit of a₀, which amounts approximately to 3 bisections; and at each step we have to solve the system (5.1) with accuracy no less than the accuracy of a₀. Of course, this method is very inefficient and allows us to get only an approximation to the desired step-like solution, with the width of well up to 50. These intermediate solutions are also quite interesting. Continuing them from the initial line t = 0, we can see several typical regions with different oscillation modes, as shown on Fig. 6.

Fig. 6.

An intermediate solution with a wide well.

A rigorous proof of the existence of step-like solutions for system (5.1) is lacking. A fairly plausible approximation to such a solution can be obtained by taking a well large enough and continuing it by the constant. An example of such approximation is shown on Fig. 7; the structure of the rarefaction and compression waves for this solution are shown on Fig. 8. In accordance to the known asymptotic formulae, the first peak of the compression wave moves parallel to the line x = −4t, t > 0 and the rarefaction wave reaches the unit limiting value along the line x = −6t, t < 0. In these regions the behaviour of solution is determined by the soliton moving at a speed 4, and the singualr solution u = −x/(6t); recall that these are exact solutions of (3.3), (3.4), see Table 1. It is interesting that these lines can be traced down even for general regular solutions, see Fig. 4.

Fig. 7.

One of step-like solutions for H₀ = −2 and H₁ = −6. View from t < 0 and top view, with the half-lines x = −6t, t < 0 and x = −4t, t > 0.

Fig. 8.

The rarefaction and compression waves.

Step-like solutions corresponding to different values of H₀ and H₁ look more or less similar, but a quantitative difference can be observed at the level of their asymptotic behaviour. The formal asymptotic solution of (5.1) at u → 0 can be found in the form of power series

by direct substitution; it turns out that and all other coefficients are uniquely defined by the recurrence relations

Thus, the asymptotic behaviour for u → 0 is completely determined by the values of the first integrals. Equations for A₂ and B₀, and numerical experiments lead us to the following conjecture.

7. Conclusion

We have demonstrated that the KdV equation (1.1) admits solutions defined by the pair of ODEs (3.3) and (3.4). For these solutions, the following sequence of degeneracies has been described:

At each stage, the definitions become less effective, which also leads to difficulties in the numerical study. In fact, step-like solutions of (3.3) and (3.4) should be simpler than more general regular solutions (like solitons are simpler than cnoidal waves), but so far we do not know an alternative way to define them. Some open problems are: rigorous proofs of the existence of regular and separatrix solutions; study of their asymptotics and connection formulae; enhancement of numerical schemes; applications to the original Gurevich–Pitaevskii problem and generalizations for other integrable models.

Acknowledgements

I would like to thank A.B. Shabat and B.I. Suleimanov for helpful conversations concerning the Gurevich–Pitaevskii problems. This work was supported in part by the Simons Foundation.