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Jun 27, 2018
06/18

by
Maria Shubina

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In this paper we investigate the one-dimensional parabolic-parabolic Patlak-Keller-Segel model of chemotaxis. For the case when the diffusion coefficient of chemical substance is equal to two, in terms of travelling wave variables the reduced system appears integrable and allows the analytical solution. We obtain the exact soliton solutions, one of which is exactly the one-soliton solution of the Korteweg-de Vries equation.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1504.07083

3
3.0

Jun 30, 2018
06/18

by
Yu Hou; Engui Fan

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This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through studying an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas....

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1406.6359

3
3.0

Jun 29, 2018
06/18

by
Jan L. Cieśliński; Artur Kobus

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It is well known that in some cases the spectral parameter has a group interpretation. We discuss in detail the case of Gauss-Codazzi equations for isothermic surfaces immersed in $E^3$. The algebra of Lie point symmetries is 4-dimensional and all these symmetries are also symmetries of the Gaus-Weingarten equations (which can be considered as so(3)-valued non-parametric linear problem). In order to obtain a non-removable spectral parameter one has to consider so(4,1)-valued linear problem...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1611.10148

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5.0

Jun 29, 2018
06/18

by
Yoshimasa Matsuno

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We propose a multi-component generalization of the modified short pulse (SP) equation which was derived recently as a reduction of Feng's two-component SP equation. Above all, we address the two-component system in depth. We obtain the Lax pair, an infinite nember of conservation laws and multisoliton solutions for the system, demonstrating its integrability. Subsequently, we show that the two-component system exhibits cusp solitons and breathers for which the detailed analysis is performed....

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1607.01079

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4.0

Jun 29, 2018
06/18

by
Supriya Mukherjee; A. Ghose Choudhury; Partha Guha

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We adopt the Chiellini integrability method to find the solutions of various generalizations of the damped Milne-Pinney equations. In particular, we find the solution of the damped Ermakov-Painlev\'e II equation and generalized dissipative Milne-Pinney equation.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1603.08747

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4.0

Jun 29, 2018
06/18

by
Yujian Ye; Zhihui Li; Shoufeng Shen; Chunxia Li

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In this letter, a new generalized matrix spectral problem of Dirac type associated with the super Lie algebra $\mathcal{B}(0,1)$ is proposed and its corresponding super integrable hierarchy is constructed.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1604.03728

3
3.0

Jun 30, 2018
06/18

by
H. Aratyn; J. F. Gomes; D. V. Ruy; A. H. Zimerman

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Exact rational solutions of the generalized Hunter-Saxton equation are obtained using Pad\'e approximant approach for the traveling-wave and self-similarity reduction. A larger class of algebraic solutions are also obtained by extending a range of parameters within the solutions obtained by this approach.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1403.1831

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3.0

Jun 29, 2018
06/18

by
Adam Hlaváč; Michal Marvan

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For the constant astigmatism equation, we construct a system of nonlocal conservation laws (an abelian covering) closed under the reciprocal transformations. We give functionally independent potentials modulo a Wronskian type relation.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1602.06861

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3.0

Jun 28, 2018
06/18

by
Alexander V Mikhailov

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We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schrodinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model, principal chiral field model as well as with differential-difference integrable systems (including the Toda lattice and differential-difference Heisenberg chain) and integrable partial difference systems. We show that there exists a transformation which formally...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1512.07664

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Jun 28, 2018
06/18

by
Nikolay K. Vitanov; Zlatinka I. Dimitrova; Kaloyan N. Vitanov

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We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of Weierstrass and Jacobi. We show that for this case the studied class of nonlinear partial differential...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1507.04722

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9.0

Jun 28, 2018
06/18

by
M. Boiti; F. Pempinelli; A. K. Pogrebkov

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Properties of the Cauchy--Jost (known also as Cauchy--Baker--Akhiezer) function of the KPII equation are described. By means of the $\bar\partial$-problem for this function it is shown that all equations of the KPII hierarchy are given in a compact and explicit form, including equations on the Cauchy--Jost function itself, time evolutions of the Jost solutions and evolutions of the potential of the heat equation.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1508.02229

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13

Jun 28, 2018
06/18

by
Nicoleta-Corina Babalic; A. S. Carstea

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We present two integrable discretisations of a general differential-difference bicomponent Volterra system. The results are obtained by discretising directly the corresponding Hirota bilinear equations in two different ways. Multisoliton solutions are presented together with a new discrete form of Lotka-Volterra equation obtained by an alternative bilinearisation.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1508.04661

9
9.0

Jun 30, 2018
06/18

by
A. S. Carstea

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Hirota bilinear form and multisoliton solution for semidiscrete and fully discrete (difference-difference) versions of supersymmetric KdV equation found by Xue, Levi and Liu [1] is presented. The solitonic interaction term displays a fermionic dressing factor as in the continuous supersymmetric case. Using bilinear equations it is shown also that there can be constructed a new integrable semidiscrete (and fully discrete) version of supersymmetric KdV which has simpler bilinear form but more...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1412.1334

3
3.0

Jun 29, 2018
06/18

by
Gregorio Benincasa; Rod Halburd

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The anti-self-dual Yang-Mills equations are known to have reductions to many integrable differential equations. A general B\"acklund transformation (BT) for the ASDYM equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1601.03102

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7.0

Jun 29, 2018
06/18

by
Francesco Giglio; Giulio Landolfi; Antonio Moro

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Inspired by the recent developments in the study of the thermodynamics of van der Waals fluids via the theory of nonlinear conservation laws and the description of phase transitions in terms of classical (dissipative) shock waves, we propose a novel approach to the construction of multi-parameter generalisations of the van der Waals model. The theory of integrable nonlinear conservation laws still represents the inspiring framework. Starting from a macroscopic approach, a four parameter family...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1602.03975

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4.0

Jun 29, 2018
06/18

by
Jia-Liang Ji; Zuo-Nong Zhu

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It is well known that the nonlinear Schr\"odinger (NLS) equation is a very important integrable equation. Ablowitz and Musslimani introduced and investigated an integrable nonlocal NLS equation through inverse scattering transform. Very recently, we proposed an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can also be found in a paper of Ablowitz and Musslimani. We have constructed the Darboux transformation and soliton solutions for the nonlocal mKdV equation. In...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1603.03994

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5.0

Jun 30, 2018
06/18

by
Ognyan Christov; Georgi Georgiev

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In this paper we study the equation $$ w^{(4)} = 5 w" (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma, $$ which is one of the higher-order Painlev\'e equations (i.e., equations in the polynomial class having the Painlev\'e property). Like the classical Painlev\'e equations, this equation admits a Hamiltonian formulation, B\"acklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1412.2867

3
3.0

Jun 30, 2018
06/18

by
S. Mamba; M. K. Folly-Gbetoula; A. H. Kara

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We derive a method for finding Lie Symmetries for third-order difference equations. We use these symmetries to reduce the order of the difference equations and hence obtain the solutions of some third-order difference equations. We also introduce a technique for obtaining their first integrals.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1701.06560

3
3.0

Jun 29, 2018
06/18

by
Sotiris Konstantinou-Rizos; Alexander V. Mikhailov

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We construct a noncommutative (Grassmann) extension of the well known Adler Yang-Baxter map. It satisfies the Yang-Baxter equation, it is reversible and birational. Our extension preserves all the properties of the original map except the involutivity.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1602.01714

4
4.0

Jun 29, 2018
06/18

by
Andrew N. W. Hone; Vladimir Novikov; Jing Ping Wang

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A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1602.03431

3
3.0

Jun 29, 2018
06/18

by
A. Ghose Choudhury; Partha Guha

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Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Li\'enard equation. By combining the Chiellini condition for integrability and Jacobi's Last Multiplier the Lagrangian and Hamiltonian of the Li\'enard equation is derived. We also show that the Kukles equation is the only equation in the Li\'enard family which satisfies both the Chiellini integrability and the Sabatini criterion for isochronicity conditions....

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1608.02319

3
3.0

Jun 29, 2018
06/18

by
G. Gubbiotti; R. I. Yamilov

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In this paper we prove that the trapezoidal $H^{4}$ and the $H^{6}$ families of quad-equations are Darboux integrable systems. This result sheds light on the fact that such equations are linearizable as it was proved using the Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic entropy, symmetries and linearization for quad equations consistent on the cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with some suggestions on how first integrals can...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1608.03506

3
3.0

Jun 29, 2018
06/18

by
N. I. Spano; A. L. Retore; J. F. Gomes; A. R. Aguirre; A. H. Zimerman

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In this paper we obtain a general expression for the n-defect matrix for the sinh-Gordon model. This in turn generate the general B\"acklund transformations (BT) for a system with $n$ type-I defects, through a gauge transformation.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1610.01856

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4.0

Jun 28, 2018
06/18

by
Ivan A. Bizyaev; Alexey V. Borisov; Ivan S. Mamaev

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In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1510.00181

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5.0

Jun 30, 2018
06/18

by
Danilo V. Ruy

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Integrable mixed models have been used as a generalization of traditional integrable models. However, a map from a traditional integrable model to a mixed integrable model is not well understood yet. Here, it is studied the relation between the mKdV-Liouville hierarchy and the mKdV hierarchy by employing an extended version of the modified truncation approach. This paper shows some solutions for the mKdV-Liouville hierarchy constructed from the soliton solutions of the mKdV hierarchy. The last...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1406.5775

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4.0

Jun 30, 2018
06/18

by
Ralph Willox; Madoka Hattori

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We present a discrete analogue of the so-called symmetry reduced or `constrained' KP hierarchy. As a result we obtain integrable discretisations, in both space and time, of some well-known continuous integrable systems such as the nonlinear Schroedinger equation, the Broer-Kaup equation and the Yajima-Oikawa system, together with their Lax pairs. It will be shown that these discretisations also give rise to a discrete description of the entire hierarchy of associated integrable systems. The...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1406.5828

4
4.0

Jun 29, 2018
06/18

by
Hidetomo Nagai

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We propose an ultradiscrete permanent solution to the ultradiscrete KP equation. The ultradiscrete permanent is an ultradiscrete analogue of the usual permanent. The elements on this ultradiscrete permanent solution are required some additional relations other than the ultradiscrete dispersion relation. We confirm the solution satisfying these relations and propose some explicit examples of the solution.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1611.09081

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Jun 27, 2018
06/18

by
Min Li; Tao Xu; Dexin Meng

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In this paper, via the generalized Darboux transformation, rational soliton solutions are derived for the parity-time-symmetric nonlocal nonlinear Schr\"{o}dinger (NLS) model with the defocusing-type nonlinearity. We find that the first-order solution can exhibit the elastic interactions of rational antidark-antidark, dark-antidark, and antidark-dark soliton pairs on a continuous wave background, but there is no phase shift for the interacting solitons. Also, we discuss the degenerate case...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1503.02254

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10.0

Jun 27, 2018
06/18

by
Vladimir Sokolov

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We consider second order differential operators $P$ with polynomial coefficients that preserve the vector space $V_k$ of polynomials of degrees not greater then $k$. We assume that the metric associated with the symbol of $P$ is flat and that the operator $P$ is potential. In the case of two independent variables we obtain some classification results and find polynomial forms for the elliptic $A_2$ and $G_2$ Calogero-Moser Hamiltonians and for the elliptic Inosemtsev model.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1503.05185

7
7.0

Jun 29, 2018
06/18

by
Maria Shubina

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In this paper we study the Painlev\'e analysis for two models of chemotaxis. We find that in some cases the reductions of these models in terms of travelling wave variable allow exact analytical solutions.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1607.00349

4
4.0

Jun 29, 2018
06/18

by
Ivan A. Bizyaev; Alexey V. Borisov; Ivan S. Mamaev

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This paper provides a detailed description of various reduction schemes in rigid body dynamics. Analysis of one of such nontrivial reductions makes it possible to order the cases already found and to obtain new generalizations of the Kovalevskaya case to e(3). We note that the above reduction allows one to obtain in a natural way some singular additive terms which were proposed earlier by D.N. Goryachev.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1607.07982

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6.0

Jun 30, 2018
06/18

by
Zhiwei Wu; Jingsong He

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We generate hierarchies of derivative nonlinear Schr\"odinger-type equations and their nonlocal extensions from Lie algebra splittings and automorphisms. This provides an algebraic explanation of some known reductions and newly established nonlocal reductions in integrable systems.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1704.02070

5
5.0

Jun 30, 2018
06/18

by
O. I. Morozov; A. Sergyeyev

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We consider the four-dimensional integrable Martinez Alonso--Shabat equation, and list three integrable three-dimensional reductions thereof. We also present a four-dimensional integrable modified Martinez Alonso--Shabat equation together with its Lax pair. We also construct an infinite hierarchy of commuting nonlocal symmetries (and not just the shadows, as it is usually the case in the literature) for the Martinez Alonso--Shabat equation.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1401.7942

4
4.0

Jun 30, 2018
06/18

by
Gui Mu; Zhenyun Qin; Roger Grimshaw

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General higher order rogue waves of a vector nonlinear Schrodinger equation (Manakov system) are derived using a Darboux-dressing transformation with an asymptotic expansion method. The Nth order semi-rational solutions containing 3N free parameters are expressed in separation of variables form. These solutions exhibit rogue waves on a multisoliton background. They demonstrate that the structure of rogue waves in this two-component system is richer than that in a one-component system. The study...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1404.2988

3
3.0

Jun 28, 2018
06/18

by
Jian Xu; Qiaozhen Zhu; Engui Fan

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We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the Sasa-Satsuma(SS) equation on the finite interval. Assume that the solution existes, we show that this solution can be expressed in terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$, $S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1512.06625

3
3.0

Jun 29, 2018
06/18

by
Yoko Shigyo

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We study the series expansion of the tau function of the BKP hierarchy applying the addition formulae of the BKP hierarchy. Any formal power series can be expanded in terms of Schur functions. It is known that, under the condition $\tau(x)\neq0$, a formal power series $\tau(x)$ is a solution of the KP hierarchy if and only if its coefficients of Schur function expansion are given by the so called Giambelli type formula. A similar result is known for the BKP hierarchy with respect to Schur's...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1601.02083

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4.0

Jun 29, 2018
06/18

by
QiLao Zha

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Explicit multi-soliton and multi-peakon solutions of the Dullin-Gottwald-Holm equation are constructed via Darboux transformation and direct computation, respectively. To this end we first map the Dullin-Gottwald-Holm equation to a negative order KdV equation by a reciprocal transformation. Then we use the Darboux matrix approach to derive multi-soliton solutions of the Dullin-Gottwald-Holm equation from the solutions of the negative order KdV equation. Finally, we find multi-peakon solutions...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1610.07051

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5.0

Jun 29, 2018
06/18

by
Nikolai A. Kudryashov; Dmitry I. Sinelshchikov

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Li\'enard-type equations are used for the description of various phenomena in physics and other fields of science. Here we find a new family of the Li\'enard-type equations which admits a non-standard autonomous Lagrangian. As a by-product we obtain autonomous first integrals for each member of this family of equations. We also show that some of the previously known conditions for the existence of a non-standard Lagrangian for the Li\'enard-type equations follow from the linearizability of the...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1608.04926

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3.0

Jun 28, 2018
06/18

by
Avinash Khare; Avadh Saxena

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A $PT$-symmetric dimer is a two-site nonlinear oscillator or a nonlinear Schr\"odinger dimer where one site loses and the other site gains energy at the same rate. We present a wide class of integrable oscillator type dimers whose Hamiltonian is of arbitrary even order. Further, we also present a wide class of integrable and superintegrable nonlinear Schr\"odinger type dimers where again the Hamiltonian is of arbitrary even order.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1510.00446

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4.0

Jun 30, 2018
06/18

by
V. P. Kotlarov

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This paper contains first results on the finite-gap integration of the Sine-Gordon equation. They were published on Russian in 1976. The papers \cite{Koz}, \cite{KK}, \cite{KK02} have been rewritten in the English language with small modifications for a convenience. Such a translation was made due to requests of some interested readers. In those papers, the method of constructing of the finite-gap solutions of the equation $u_{tt}-u_{xx}+\sin u=0$ was proposed. The explicit formulae were...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1401.4410

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3.0

Jun 30, 2018
06/18

by
G. Shaikhova; K. Yesmakhanova; G. Mamyrbekova; R. Myrzakulov

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In this paper, we construct a Darboux transformation (DT) of the (2+1)-dimensional Schr\"odinger-Maxwell-Bloch equation (SMBE) which is integrable by the Inverse Scattering Method. Using this DT, the one-soliton solution and periodic solution are obtained from the "seed" solutions.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1402.4669

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3.0

Jun 30, 2018
06/18

by
Claude M. Viallet

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We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1408.6161

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Jun 30, 2018
06/18

by
R. Gladwin Pradeep; V. K. Chandrasekar; R. Mohanasubha; M. Senthilvelan; M. Lakshmanan

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We identify contact transformations which linearize the given equations in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The identified contact transformations are not of Cole-Hopf type and are \emph {new} to the literature. The linearization of Abel chain of equations is also demonstrated explicitly for the first time. The contact transformations can be utilized to derive dynamical symmetries of the associated nonlinear ODEs. The...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1410.4704

41
41

Jun 27, 2018
06/18

by
Yoshimasa Matsuno

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A novel method is developed for extending the Green-Naghdi (GN) shallow-water model equation to the general system which incorporates the arbitrary higher-order dispersive effects. As an illustrative example, we derive a model equation which is accurate to the fourth power of the shallowness parameter while preserving the full nonlinearity of the GN equation, and obtain its solitary wave solutions by means of a singular perturbation analysis. We show that the extended GN equations have the same...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1503.08520

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3.0

Jun 30, 2018
06/18

by
Yoichi Nakata

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We propose a method to solve the initial value problem for the ultradiscrete Somos-4 and Somos-5 equations by expressing terms in the equations as convex polygons and regarding max-plus algebras as those on polygons.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1701.04262

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5.0

Jun 30, 2018
06/18

by
Danda Zhang; Da-jun Zhang

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In the paper we derive rational solutions for the lattice potential modified Korteweg-de Vries equation, and Q2, Q1($\delta$), H3($\delta$), H2 and H1 in the Adler-Bobenko-Suris list. B\"acklund transformations between these lattice equations are used. All these rational solutions are related to a unified $\tau$ function in Casoratian form which obeys a bilinear superposition formula.

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1702.01266

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3.0

Jun 28, 2018
06/18

by
M. A. Reyes; D. Gutierrez-Ruiz; S. C. Mancas; H. C. Rosu

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We present an approach to the bright soliton solution of the NLS equation from the standpoint of introducing a constant potential term in the equation. We discuss a `nongauge' bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sech^p solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg-de Vries and...

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1510.02994

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6.0

Jun 30, 2018
06/18

by
Zakhar V. Makridin; Maxim V. Pavlov

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In this paper we introduce a new property of two-dimensional integrable systems -- existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional commuting flows. Infinitely many three-dimensional local conservation laws for the Korteweg de Vries pair of commuting flows and for the Benney commuting hydrodynamic chains are constructed. As a by-product we established a new method for computation of local conservation laws for three-dimensional...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1704.04005

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Jun 28, 2018
06/18

by
M. S. Abdel Latif

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In this paper, we show that the improved (G'/G)- expansion method is equivalent to the tanh method and gives the same exact solutions of nonlinear partial differential equations.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences

Source: http://arxiv.org/abs/1506.06025

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10.0

Jun 30, 2018
06/18

by
Timofey Zolkin; Sergei Nagaitsev; Viatcheslav Danilov

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Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to a Twist map, with a rotation number, constant along the phase trajectory. In this letter, we propose a succinct expression to determine the rotation number and present two examples. Similar to the period of the bounded motion in Hamiltonian systems, the...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems

Source: http://arxiv.org/abs/1704.03077