Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions

The nonisospectral method to generate solvable nonlinear evolution equations was proposed in the seminal paper [3] and actively used in a wide range of papers (see, for instance, various references in the recent papers [18,33,34]). In particular, this approach allows to generate physically interesting nonlinear equations with external potentials [4,5]. Such equations are used to describe deep water and plasma waves, waves in non-uniform media, light pulses, and energy transport (see [7,30,33] and references therein).

In the paper we assume that the spectral parameter λ depends on both variables x and t. In this way we construct solvable generalizations of the matrix versions of the coupled nonlinear Schrödinger (CNLS), KdV and MKdV equations. Matrix versions are of interest (see, for instance, [6]) and include, in particular, scalar cases and multi-component cases. We construct auxiliary linear systems for our matrix (and scalar) cases and some equations seem to be new in the scalar cases too.

After that we apply to the constructed equations GBDT, which is a version of the Bäcklund-Darboux transformation developed by the author in [20]- [27]. The Bäcklund-Darboux transformation is a well-known and fruitful tool to construct explicit solutions of the integrable equations and some classical linear equations as well. Some important versions of the Bäcklund-Darboux transformation one can find in [8,9,12,14,16,17,32]. The GBDT version of the iterated Bäcklund-Darboux transformation is of a rather general nature and provides simple algebraic formulas based on some system theoretical and matrix identities results. The parameter matrices, which are used in GBDT, have an arbitrary Jordan structure, while diagonal matrices (with eigenvalues of the auxiliary spectral problems as the entries) are mostly used in other approaches.

Examples are considered in a more detailed way.

2 Nonisospectral equations

Recall the coupled nonlinear Schrödinger equation (CNLS) of the form [11]:

We shall consider the matrix version of (2.1), where v 1 and v 2 are m 1 × m 2 and m 2 × m 1 (m 1 , m 2 ≥ 1) matrix functions, respectively. Auxiliary linear systems for CNLS are given by the formulae

Here we have

where λ is the independent of x and t spectral parameter,

)

and I k is the k × k identity matrix. We assume in this section that G and F are continuous together with their first derivatives. Then the compatibility condition for systems (2.2) can be written down in the zero curvature equation form

It can be checked directly that (2.1) is equivalent to the compatibility condition (2.7). In other words, equation (2.1) can be presented in the zero curvature form (2.7). (See [10] on the historical details about zero curvature representation of the integrable equations.) It follows from (2.7) that (2.1) can be presented in the form:

-j ξ t (x, t) + ijξ xx (x, t) + 2ijξ(x, t) 3 = 0.

(2.8)

Indeed, the left hand side in (2.8) coincides with the left hand side in (2.7), while the coefficients at the degrees λ k (k > 0) on the left hand side in (2.7) turn to zero. It is easy to calculate that in the nonisospectral case (for λ depending on x and t) the zero curvature equation (2.7) is equivalent to

to derive from (2.4), (2.9) and (2.10)

Thus we obtain the following proposition Proposition 2.1 The CNLS with external potential

The problem of similarity transformations [15,19] is of interest here. When b = b, using λ(x, t) as in (2.13) one can construct an integrable nonlinear Schrödinger equation (NLS) with external potential [26], however the substi-

turns it (see [19], Section 6, Example 4) into the classical cubic NLS. In a quite similar way the substitution (2.15) and equalities

(2.17)

The case b = b is more interesting from that point of view, though GBDT for the equation (2.14) (b = b) can be applied to construct new solutions of (2.1) and proves therefore useful too.

The matrix KdV equation can be written as:

where v is an p × p matrix function. Equation (2.18) admits zero curvature representation (2.7), where polynomials G and F of the form (2.3) are defined via the m × m (m = 2p) coefficients

(2.20) Now, substitute (2.20) by the equalities

where g and f are scalar functions. One can easily take into account corresponding changes in (2.7) and obtain our next proposition.

Notice that equation (2.22), where p = 1 (scalar case) and h = 0, was treated using the homogeneous balance principle in [31]. When (2.23) holds and g = 1, one can put f = -(t + b) -1 /12. The corresponding equation

appeared in [1] and its subcase b = 0 is a well-known cylindrical KdV [4].

Introduce G and F by the equalities

where

Then representation (2.7) is equivalent to the equation

. (2.27) Some other generalizations of MKdV one can find in [2,33].

GBDT (nonisospectral case) for systems with rational dependence on the spectral parameter have been introduced in [21]. Here we shall need a reduction of the theorem in [21] (Section 2, p. 1253) fo

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