Explicit solutions of some equations and systems of mathematical physics

This paper deals at first with a fully integrable evolution system of nonlinear partial differential equations (PDEs) which is a generalization of the classical Heisenberg ferromagnet equation. Then the scalar variant of this system is considered. Looking for solutions of special form, the problem of finding explicit solutions of the above-mentioned equations is reduced to the global solvability of overdetermined real-valued systems of nonlinear PDEs. In many cases particular solutions which are not solitons are expressed by classical functions including some special ones as Jacobi elliptic functions, Legendre elliptic functions, and Weierstrass normal elliptic integrals. A geometrical visualization of several solutions is also proposed.


Introduction
This paper deals at first with a 2 × 2 fully integrable evolution system of nonlinear partial differential equations. It is a generalization of the classical Heisenberg ferromagnet equation, where the solution S is the unit spin vector (see [2]). A relatively new integrable model was proposed in [6], and two types of soliton solutions, namely quadruplet and doublet solution, were found. In [12] other particular solutions of the same system were constructed. They turn out to be either soliton type ones or quasi-rational ones. In both cases the Zakharov-Shabat dressing method was applied (for this method, see for example [13][14][15]). The inverse scattering method is a good approach for finding exact solutions of different equations of mathematical physics, too. The above-mentioned methods were intensively used in the last 60 years, and due to them a big progress in the investigation of nonlinear evolution equations and systems was made.
We propose here another classical method on the subject which is applied to the evolution system under consideration (see Sect. 2, Eq. (1)) as well to its scalar variant. The latter is a generalization of the derivative Schrödinger equation studied in [8] from the point of view of the explicit solutions. Some results on the Cauchy problem iu t + u xx + F(u,ū, u xūx ) = 0 can be found in [3]. Here we look for solutions of special form (Ansatz) reducing this way our problem to the global solvability of (over) determined real-valued systems of PDEs. Inverse scattering theory is avoided but rather complicated systems of real PDEs could appear. On the other hand, this simple and direct approach guarantees in several special cases the expression of the corresponding solutions by some well-studied special functions as Jacobi elliptic functions, Legendre elliptic functions, and Weierstrass normal elliptic integrals of first, second, and third kinds (see for details [4,5,7]). As one can guess thereby different kinds of non-soliton solutions can be constructed. Theorem 1 of our paper for (1) can be compared with some results of [12] where it is supposed additionally that u(±∞, t) = 0, v(±∞, t) = 1. Certainly, the solution (u, v) in Theorem 1(b) does not satisfy the latter assumption and is not soliton. Some of the stationary soliton solutions from [12] are (u = 0, v = e iϕ(x) , ϕ(±∞) ∈ 2πZ), while in Theorem 1(a) the stationary solution is (cos k 0 e iϕ(x) , sin k 0 e iϕ(x) ), ϕ being arbitrary real-valued function, k 0 = const. For a special choice of the rational amplitudes f 1 , f 2 in Anzatz (2) the corresponding solution can blow up at some curve in the plane 0xt. Similar effect is found in [12] on the level of rather complicated example (117), (118) (quasi-rational solution). The solutions of the scalar equation (8) in [6,12] are of soliton type (see [6] In what follows, we propose geometrical visualization of u. For example in Theorem 2, 2) the amplitude f is located between two parallel oblique asymptotes and can be expressed by some Jacobi elliptic functions. In Theorem 2, 3) the amplitude f (k) = cos k, k = k(x) and k can be written by Weierstrass ℘ and zeta functions. Geometrically, k(x) is possibly periodic cuspon or soliton-cuspon. In the papers devoted to Eq. (8) mainly soliton solutions are proposed due to the used tools-inverse scattering and dressing methods. The appearance of autonomous ordinary differential equations satisfied by Ansatz (9) enables us to enlarge the classes of the particular solutions, to express them by some special functions, and to propose geometrical interpretation. The soliton solutions for (8) in the literature we know are usually written by elementary functions, i.e., exponents, logarithms, and trigonometric ones (see [6,12], and others).
We do not know papers on the subject in which solutions of other form are found explicitly and studied geometrically. We do not discuss the possible applications of those solutions in physics being concentrated on the mathematical part of the study of the arising differential equations. The results from Theorem 2 are illustrated by four figures.

Formulation of the main results
1. Consider the fully integrable nonlinear evolution system Usually, the additional condition |u| 2 + |v| 2 = 1 is imposed on (1). In vector form the Heisenberg ferromagnet equation is given by S t = S × S xx , S(x, t) ∈ R 3 , |S| = 1.
We look for a solution of (1) having the form where f 1 (k), f 2 (k), k, ϕ are real-valued smooth functions. Putting (2) into (1), doing the corresponding calculations, and splitting the real and imaginary parts of the expressions, we get the following overdetermined system of four PDEs that should be satisfied by k, ϕ: where g(k) = f 2 1 + f 2 2 . The simplest case to system (3)-(6) is the following case: (A) f 1 = cos k, f 2 (k) = sin k ⇒ g ≡ 1, i.e., g (k) = 0. Therefore, |u| 2 + |v| 2 = 1. In case (A) system (3)-(6) can be reduced to The solutions in Theorem 1 are globally defined in R 2 . For some rational functions f 1 , f 2 smooth in R 2 , we can find solutions k that blow up along some curves in the plane. The corresponding example is given at the end of the proof of Theorem 1.
2. Our next step is to study system (1) for v ≡ 0, i.e., the nonlinear evolution equation of Schrödinger type, namely u is not obliged to satisfy the condition |u| = 1.
As in the previous system, we look for its solution of the form f (k), k, ϕ being again real-valued smooth functions. Putting (9) into (8) and splitting the real and imaginary parts of the corresponding expression, we conclude that k, ϕ satisfy the following nonlinear system of PDEs: We shall construct solutions of (8) written in the form of (9) in several different cases. They are formulated in what follows.

Theorem 2 Consider the nonlinear evolution scalar Eq. (8).
Then we give its solutions of the form (9) in the following four cases.
where γ ∈ R 1 , α j with |α j | < 1 are arbitrary complex numbers, and the complex-valued function z(x) = 0 everywhere, satisfy (8). Let A, B ∈ C 2 , A, B -real-valued and |A(x)| > 0 everywhere. Then (12) takes the trigonometric form 2) Suppose that f ≡ 1 and k, ϕ are linear functions with respect to (t, x). Then there exists a smooth solution f of (10), (11) possessing two oblique parallel each to other asymptotes and f is located between them. Moreover, f can be expressed by the Legendre elliptic functions as well as by some Jacobi elliptic functions.
3) Suppose that f (k) = cos k, k = k(x), and ϕ = t + ϕ 1 (x) satisfy (10)- (11). Then there is a simple link between k(x), ϕ 1 (x), while k(x) satisfies a second order autonomous ODE and can be expressed by Weierstrass normal elliptic integrals of first and second kind, i.e., by ℘ and zeta functions. Under several conditions k turns out to be periodic cuspon or solitoncuspon. 4

be expressed explicitly by k(x), while k(x) is expressed by the Weierstrass normal elliptic integrals of first and second kinds.
There are other interesting cases as for example f (k) = sin k, f (k) = P m (k), P m being realvalued polynomial of k of order m. We left them to the reader, as they can be studied in the same way as in Theorem 2.
In a similar way one can study the system under the condition |u| 2 -|v| 2 = 1 looking for a solution of the form (2).
2. We shall prove now Theorem 2, the proof being divided into several steps depending on the cases.
According to (10), f satisfies the ODE We equip (14) with the Cauchy data f (k 0 ) = f 0 , f (k 0 ) = f 0 . The standard change in (14) df df (p 2 ) and the substitution p 2 = q(f ) transform (14) into the linear first order ODE with respect to q: Then q can be written as After some calculations we obtain for f (k) the relation df Here we apply some technique from [1]. We shall study the case with the sign + in front of √ q as the other case is similar. Thus, Put One can guess that the function z = F(f ) possesses two oblique parallel each to other asymptotes Therefore, f = F -1 (kk 0 ), k ∈ R 1 , f (k 0 ) = f 0 and f has the following oblique asymptotes: Fig. 1).
The latter integral is of the type (31) and as in case 3 can be written by the Weierstrass elliptic integrals 1), 2).