Journal of Nonlinear Mathematical Physics

Volume 17, Issue 4, December 2010
Letter to Editor

1. Peakons Arising as Particle Paths Beneath Small-Amplitude Water Waves in Constant Vorticity Flows

Delia Ionescu-Kruse
Pages: 415 - 422
We present a new kind of particle path in constant vorticity water of finite depth, within the framework of small-amplitude waves.
Letter to Editor

2. Some Examples of Algebraic Geodesics on Quadrics. II

A. M. Perelomov
Pages: 423 - 428
In this note we give new examples of algebraic geodesics on some two-dimensional quadrics, namely, on ellipsoids, one-sheet hyperboloids, and hyperbolic paraboloids. It appears that in all considered cases, such geodesics are rational space curves.
Research Article

3. Noncommutative Hypergeometric and Basic Hypergeometric Equations

Alessandro Conflitti, Michael J. Schlosser
Pages: 429 - 443
Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100(14) (2003) 8138–8141] considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give...
Research Article

4. Explicit Soliton Asymptotics for the Nonlinear Schrödinger Equation on the Half-Line

K. Kalimeris
Pages: 445 - 452
There exists a particular class of boundary value problems for integrable nonlinear evolution equations formulated on the half-line, called linearizable. For this class of boundary value problems, the Fokas method yields a formalism for the solution of the associated initial-boundary value problem, which...
Research Article

5. Bi-Hamiltonian Representation, Symmetries and Integrals of Mixed Heavenly and Husain Systems

M. B. Sheftel, D. Yazici
Pages: 453 - 484
In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here...
Research Article

6. The Quantization of a Fourth-Order Equation without a Second-Order Lagrangian

M. C. Nucci, P. G. L. Leach
Pages: 485 - 490
We present an equation of the fourth-order which does not possess a second-order Lagrangian and demonstrate by means of the method of reduction of order that one can obtain a first-order Lagrangian for it. This opens the way to quantization through the construction of an Hamiltonian which is suitable...
Research Article

7. Lax Pair, Binary Darboux Transformation and New Grammian Solutions of Nonisospectral Kadomtsev–Petviashvili Equation with the Two-Singular-Manifold Method

Shou-Fu Tian, Hong-Qing Zhang
Pages: 491 - 502
In this letter, the two-singular-manifold method is applied to the (2+1)-dimensional nonisospectral Kadomtsev–Petviashvili equation with two Painlevé expansion branches to determine auto-Bäcklund transformation, Lax pairs and Darboux transformation. Based on the two obtained Lax pairs, the binary Darboux...
Research Article

8. Nonlinear Stability Analysis of the Emden–Fowler Equation

C. G. Böhmer, T. Harko
Pages: 503 - 516
In this paper, we qualitatively study radial solutions of the semilinear elliptic equation Δu+un = 0 with u(0) = 1 and u′(0) = 0 on the positive real line, called the Emden–Fowler or Lane–Emden equation. This equation is of great importance in Newtonian astrophysics and the constant n is called the polytropic...
Research Article

9. Generalized, Master and Nonlocal Symmetries of Certain Deformed Nonlinear Partial Differential Equations

R. Sahadevan, L. Nalinidevi
Pages: 517 - 538
It is shown that the deformed Nonlinear Schrödinger (NLS), Hirota and AKNS equations with (1 + 1) dimension admit infinitely many generalized (nonpoint) symmetries and polynomial conserved quantities, master symmetries and recursion operator ensuring their complete integrability. Also shown that each...
Research Article

10. Existence of Natural and Conformally Invariant Quantizations of Arbitrary Symbols

P. Mathonet, F. Radoux
Pages: 539 - 556
A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection. In general, there is no natural quantization procedure, that is, spaces of...
Research Article

11. Propagation of Twist Solitons in Fully Inhomogeneous DNA Chains

Mariano Cadoni, Roberto de Leo, Sergio Demelio, Giuseppe Gaeta
Pages: 557 - 569
In the framework of a recently introduced model of DNA torsional dynamics, we argued — on the basis of perturbative considerations — that an inhomogeneous DNA chain could support long-lived soliton-type excitations due to the peculiar geometric structure of DNA and the effect of this on nonlinear torsional...
Research Article

12. Optimal Solution for the Viscous Modified Camassa–Holm Equation

Anna Gao, Chunyu Shen
Pages: 571 - 589
In this paper, we study the optimal control problem for the viscous modified Camassa–Holm equation. We first prove the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Furthermore, the existence of an optimal solution to the viscous modified...
Research Article

13. Menelaus Relation, Hirota–Miwa Equation and Fay's Trisecant Formula are Associativity Equations

B. G. Konopelchenko
Pages: 591 - 603
It is shown that the celebrated Menelaus relation, Hirota–Miwa bilinear equation for KP hierarchy and Fay's trisecant formula similar to the WDVV equation are associativity conditions for structure constants of certain three-dimensional quasi-algebra.