Numerical study of the KP equation for non-periodic waves
The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions in this paper. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. In this paper, we study the initial value problem of the KP equation with V- and X-shape initial waves consisting of two distinct line-solitons by means of the direct numerical simulation. We then show that the solution converges asymptotically to some of those exact soliton solutions. The convergence is in a locally defined -sense. The initial wave patterns considered in this paper are related to the rogue waves generated by nonlinear wave interactions in shallow water wave problem.
keywords:Kadomtsev-Petviashvili equation, soliton solutions, chord diagrams, pseudo-spectral method, window technique
The KdV equation may be obtained in the leading order approximation of an asymptotic perturbation theory for one-dimensional nonlinear waves under the assumptions of weak nonlinearity (small amplitude) and weak dispersion (long waves). The initial value problem of the KdV equation has been extensively studied by means of the method of inverse scattering transform (IST). It is then well-known that a general initial data decaying rapidly for large spatial variable evolves into a sum of individual solitons and some weak dispersive wave trains separated away from solitons (see for examples, AS:81 (); N:85 (); NMPZ:84 (); Wh:74 ()).
In 1970, Kadomtsev and Petviashvili KP:70 () proposed a two-dimensional dispersive wave equation to study the stability of one soliton solution of the KdV equation under the influence of weak transversal perturbations. This equation is now referred to as the KP equation, and considered to be a prototype of the integrable nonlinear dispersive wave equations in two dimensions. The KP equation can be also represented in the Lax form, that is, there exists a pair of linear equations associated with the eigenvalue problem and the evolution of the eigenfunctions. However, unlike the case of the KdV equation, the method of IST based on the pair of linear equations does not seem to provide a practical method for the initial value problem with non-periodic waves considered in this paper. At the present time, there is no feasible analytic method to solve the initial value problem of the KP equation with initial waves having line-solitons in the far field.
In this paper, we study this type of the initial value problem of the KP equation by means of the direct numerical simulation. In particular, we consider the following two cases of the initial waves: In the first case, the initial wave consists of two semi-infinite line-solitons forming a V-shape pattern, and in the second case, the initial wave is given by a linear combination of two infinite line-solitons forming X-shape. Those initial waves have been considered in the study of the generation of large amplitude waves in shallow water PTLO:05 (); TO:07 (); KOT:09 (). The main result of this paper is to show that the solutions of the initial value problem with those initial waves asymptotically converge to some of the exact soliton solutions found in CK:08 (); CK:09 (). This implies a separation of the (exact) soliton solution from the dispersive radiations in the similar manner as in the KdV case.
The paper is organized as follows: In Section 2, we provide a brief summary of the soliton solutions of the KP equation and the classification theorem obtained in CK:08 (); CK:09 () for those soliton solutions as a background necessary for the present study. In particular, we introduce the parametrization of each soliton solution with a chord diagram which represents a derangement of the permutation group, i.e. permutation without fixed point. In Section 3, we present several exact soliton solutions, and describe some properties of those solutions. Each of those soliton solutions has numbers of line-solitons in a far field on the two-dimensional plane, say in , and numbers of line-solitons in the far field of the opposite side, i.e. . This type of soliton solution is referred to as an -soliton solution. Here we consider those solitons with and . In Section 4, we describe the numerical scheme used in this paper, which is based on the pseudo-spectral method combined with the window technique S:05 (); TOM:08 (). The window technique is especially used to compute our non-periodic problem which is essentially an infinite domain problem. Finally, in Section 5, we present the numerical results of the initial value problems with V- and X-shape initial waves, and show that the solutions asymptotically converge to some of those exact solutions discussed in Section 3. The convergence is in the sense of locally defined -sense with the usual norm, i.e. where is a compact set which covers the main structure describing the (resonant) interactions in the solution. We also propose a method to identify an exact solution for a given initial wave with V- or X-shape pattern based on the chord diagrams introduced in the classification theory.
Here we give a brief summary of the recent result of the classification theorem for soliton solutions of the KP equation (see K:04 (); CK:08 (); CK:09 () for the details). In particular, each soliton solution is then parametrized by a chord diagram which represents a unique element of the permutation group. We use this parametrization throughout the paper.
2.1 The KP equation
The KP equation is a two-dimensional nonlinear dispersive wave equation given by
where etc. Let us express the solution in the form,
where the function is called the tau function, which plays a central role in the KP theory. In this paper, we consider a class of the solutions, where each solution can be expressed by in the Wronskian determinant form , i.e.
with for . Here the functions form a set of linearly independent solutions of the linear equations,
(The fact that (2.2) with (2.3) gives a solution of the KP equation is well-known and the proof can be found in several places, e.g. see H:04 (); CK:09 ().) The solution of those equations can be expressed in the Fourier transform,
with an appropriate contour in and the measure . In particular, we consider a finite dimensional solution with with , i.e.
Thus this type of solution is characterized by the parameters and the matrix of rank, that is, we have
Note that gives a basis of and spans an -dimensional subspace of . This means that the -matrix can be identified as a point on the real Grassmann manifold Gr (see K:04 (); CK:09 ()). More precisely, let be the set of all matrices of rank . Then Gr can be expressed as
where GL is the general linear group of rank . This is saying that other basis for any spans the same subspace. Notice here that the freedom in the -matrix with GL can be fixed by expressing in the reduced row echelon form (RREF). We then assume throughout this paper that the -matrix is in the RREF, and show that the -matrix plays a crucial role in our discussion on the asymptotic behavior of the initial value problem.
Now using the Binet-Cauchy Lemma for the determinant, the -function of (2.2) can be expressed in the form,
where is the minor of the -matrix with columns marked by , and is given by
From the formula (2.6), one can see that for a given -matrix,
if a column of has only zero elements, then the exponential term with being the column index never appear in the -function, and
if a row of has only the pivot as non-zero element, then with being the row index can be factored out from the -function, i.e. has no contribution to the solution.
We then say that an -matrix with no such cases is irreducible, because the -function with reducible matrix can be obtained by a matrix with smaller size.
We are also interested in non-singular solutions. Since the solution is given by , the non-singular solutions are obtained by imposing the non-negativity condition on the minors,
This condition is not only sufficient but also necessary for the non-singularity of the solution. We call a matrix having the condition (2.7) totally non-negative matrix.
2.2 Line-soliton solution and the notations
Let us here present the simplest solution, called one line-soliton solution, and introduce several notations to describe the solution. A line-soliton solution is obtained by a -function with two exponential terms, i.e. the case and in (2.5): With the -matrix of the form , we have
The parameter in the -matrix must be for a non-singular solution (i.e. totally non-negative -matrix), and it determines the location of the soliton solution. Since leads to a trivial solution, we consider only (i.e. irreducible -matrix) . Then the solution gives
Thus the solution is located along the line . We here emphasize that the line-soliton appears at the boundary of two regions, in each of which either or becomes the dominant exponential term, and because of this we also call this soliton -soliton solution (or soliton of -type). In general, the line-soliton solution of -type with has the following structure (sometimes we consider only locally),
with some constant . The amplitude , the wave-vector and the frequency are defined by
The direction of the wave-vector is measured in the counterclockwise from the -axis, and it is given by
with . For the parameter giving the location of the line-soliton, we also use the notation,
with in (2.8) which is determined by the -matrix and the -parameters. For multi-soliton solutions, one can only define the location of each -soliton using the asymptotic position in the -plane either or (we mainly consider the cases where the solitons are not parallel to the -axis), and we use the notation (or ) which describes the -intercept of the line determined by the wave crest of -soliton in the region (or ) at . In Figure 2.1, we illustrate an example of one line-soliton solution. In the right panel of this figure, we show a chord diagram which represents this soliton solution. Here the chord diagram indicates the permutation of the dominant exponential terms and in the -function, that is, with the ordering , dominates in , while dominates in . This representation of the line-solitons in terms of the chord diagrams is the key concept throughout the present paper. (See section 2.3 below for the precise definition of the chord diagrams.)
For each soliton solution of (2.9), the wave vector and the frequency satisfy the soliton-dispersion relation, i.e.,
The soliton velocity defined by is given by
Note in particular that , and this implies that the -component of the velocity is always positive, that is, any soliton propagates in the positive -direction. On the other hand, one should note that any small perturbation propagates in the negative -direction, that is, the -component of the group velocity is always negative. This can be seen from the dispersion relation of the KP equation for a linear wave with the wave-vector and the frequency ,
from which the group velocity of the wave is given by
This is similar to the case of the KdV equation, and we expect that asymptotically soliton separates from small radiations. Physically this implies that soliton is a supersonic wave due to its nonlinearity (recall that the velocity of shallow water wave is proportional to the square root of the water depth, and the KP equation in the form (2.1) describes the waves in the moving frame with the phase velocity in the -direction).
Stability of one-soliton solution was shown in the original paper by Kadomtsev and Petviashvii KP:70 (), and this may be stated as follows: For any and any , there exists so that if the initial wave satisfies
for some exact soliton solution, with appropriate constants and (recall ), then the stability implies that the solution satisfies
where is a circular disc with radius moving with the soliton, i.e.
with at any point on the soliton, . Here is the usual -norm of over a compact domain , i.e.
This stability implies a separation of the soliton from the dispersive radiations (non-soliton parts) as in the case of the KdV soliton. We would like to prove the similar statement for more general initial waves. However there are several difficulties for two-dimensional stability problem in general. In this paper, we will give a numerical study for some special cases where the initial waves consist of two semi-infinite line solitons with V- or X-shape.
Finally we remark that a line-soliton having the angle has an infinite speed, and of course it is beyond the assumption of the quasi-two dimensionality. We also emphasize that the structure of the solution for can be different from that in , that is, the set of asymptotic solitons in can be different from that in . This difference is a consequence of the resonant interactions among solitons as we can see throughout the paper.
2.3 Classification Theorems
Let be the pivot indices, and let be non-pivot indices for an irreducible and totally non-negative -matrix. Then the soliton solution generated by the -function with the -matrix has the following asymptotic structure:
For , there are line-solitons of -type for .
For , there are line-solitons of -type for .
Here and are determined uniquely from the -matrix.
The unique index pairings and in Theorem 2.1 have a combinatorial interpretation. Let us define the pairing map such that
where and are respectively the pivot and non-pivot indices of the -matrix. Then we have:
The pairing map is a bijection. That is, , where is the group of permutation for the index set , i.e.
Note in particular that the corresponding is the derangement, i.e. has no fixed point.
Theorem 2.2 shows that the pairing map for an -soliton solution has excedances, i.e. for , and the excedance set is the set of pivot indices of the -matrix. We represent each soliton solution with the chord diagram defined as follows:
There are marked points on a line, each of the point corresponds to the -parameter.
On the upper side of the line, there are chords (pairings), each of them connects two points on the line representing for , i.e. the excedance .
On the lower side of the line, there are chords representing for , i.e. the deficiency .
In Figure 2.2, we illustrate an example of the chord diagram which represents the derangement,
The diagram then shows that the set of excedances is , and the corresponding soliton solution consists of the asymptotic line-solutions of -, -, - and -types in and of -, -, - and -types in .
3 Exact solutions
Here we present several exact solutions generated by smaller size matrices with and . Those solutions give a fundamental structure of general solutions, and we will show that some of those solutions appear naturally as asymptotic solutions of the KP equation for certain classes of initial waves related to the rogue wave generation PTLO:05 (); TO:07 (); TOM:08 (); F:80 (). The detailed discussions and the formulae given in this section can be found in CK:09 ().
3.1 Y-shape solitons: Resonant solutions
We first discuss the resonant interaction among line-solitons, which is the most important feature of the KP equation (see e.g. M:77 (); NR:77 (); KY:80 ()). To describe resonant solutions, let us consider the -function with , that is, the -function has three exponential terms . In terms of the -function in the form (2.3), those are given by the cases and .
Let us first study the case with and , where the -matrix is given by
The parameters in the matrix are positive constants, and the positivity implies the irreducibility and the regularity of the solution. The -function is simply given by
(Note that if one of the parameters is zero, then the -function consists only two exponential terms and it gives one line-soliton solution, i.e. reducible case.) With the ordering , the solution consists of -soliton for and - and -solitons for . Taking the balance between two exponential terms in the -function, one can see that those line-solitons of - and -types are localized along the lines given in (2.9), i.e. for and where the locations are determined by the -matrix (see (2.10)),
where . The shape of solution generated by with (i.e. at three line-solitons meet at the origin) is illustrated via the contour plot in the first row of Figure 3.1.
This solution represents a resonant solution of three line-solitons, and the resonant condition is given by
which are trivially satisfied with and . The chord diagram corresponding to this soliton is shown in the right panel of the first raw in Figure 3.1, and it represents the permutation .
Let us now consider the case with and : We take the -matrix in the form,
where and are positive constants. Then the -function is given by
with for . In this case we have - and -solitons for and -soliton for , and this solution can be labeled by . Those line-solitons of - and -types are localized along the lines, with
where . In the lower figures of Figure 3.1, we illustrate the solution in this case. Notice that this figure can be obtained from -soliton in the upper figure by changing . Here the parameters and in the -matrix are chosen, so that , that is, all of those soliton solutions meet at the origin at .
3.2 and cases
Let us first discuss the case with and , that is, the -matrix is given by
where and are positive constants. The -function is simply written in the form
In this case, we have -soliton solution consisting of one line-soliton of -type for and three line-solitons of -, - and -types for . This solution is labeled by . The upper figures in Figure 3.2 shows the time-evolution of the solution of this type. We set of the -matrix, so that all four line-solitons meet at the origin at .
For , any irreducible and totally non-negative -matrix has the form,
where and are positive constants. The -function is then given by
with . This gives -soliton solution which is dual to the case of , that is, -soliton for and -, - and -solitons for . The corresponding label for this solution is given by . The lower figures in Figure 3.2 shows the time-evolution of this type. Here we set of the -matrix as
where , so that all four solitons meet at the origin at .
3.3 and cases
By a direct construction of the derangements of with two exedances (i.e. ), one can easily see that there are seven cases with irreducible and totally non-negative -matrices. Then the classification theorems imply that we have a -soliton solution associated to each of those -matrices. We here discuss all of those -soliton solutions with the same -parameters given by , and show how each -matrix chooses a particular set of line-solitons. In Figure 3.3, we illustrate the corresponding chord diagrams for all those seven cases, from which one can find the asymptotic line-solitons in each case.
From the chord diagram in Figure 3.3, one can see that the asymptotic line-solitons are given by - and -types for both . The solution in this case corresponds to the top cell of Gr, and the -matrix is given by
where are free parameters with . This is the generic solution on the maximal dimensional cell, and it is called T-type (after K:04 ()). The most important feature of this solution is the generation of four intermediate solitons forming a box at the intersection point. Those intermediate solitons are identified as -, -, - and -solitons, and they may be obtained by cutting the chord diagram of -type at the crossing points. For example, if we cut the chords of and at the crossing point, we obtain either pair of or . The first pair appears in the lower and upper edge of the box, and the second pair in the right and left edges. Figure 3.4 shows the time-evolution of the solution of this type. The parameters and in the -matrix determine the locations of solitons, their phase shifts and the on-set of the box-shaped interaction pattern: Those are given by
where is for the phase shift, for the location of -soliton for (recall the definition below (2.10), i.e. gives the -intercept of the line of the -soliton for at ), and for the on-set of the box. The phase shifts for - and -solitons are given by with
Among the soliton solutions for and , this solution is the most complicated and interesting one. As you can see below, this solution contains all other solutions as some parts of this solution, that is, as explained above, all six possible solitons (i.e. ) appear in this solution.
From the chord diagram in Figure 3.3, one can see that the asymptotic line-solitons are given by - and -solitons for and - and -solitons in . The -matrix in this case is given by
where are free parameters. Figure 3.5 illustrates an example of this solution.
The parameters in the -matrix determine the locations of the line-solitons in the following form,
The represent the locations of the -soliton for and , and the three line-solitons determine the location of the other one. We here take the parameters , so that all those four solitons meet at the origin at . Notice here that the pattern in the lower part () of the solution is the same as that of the previous one, T-type. This can be also seen by comparing the chord diagrams of those two cases.
The asymptotic line-solitons in this case are given by - and -solitons for and - and -solitons in . The -matrix has the form,
where are free parameters. Figure 3.6 illustrates the evolution of the solution of this type.
The parameters in the -matrix are related to the locations of the line-solitons,