A Riemann-Hilbert Approach for the Novikov Equation

Monvel, Anne Boutet de; Shepelsky, Dmitry; Zielinski, Lech

doi:10.3842/sigma.2016.095

Cited by 21 publications

(25 citation statements)

References 46 publications

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“…A similar situation takes place, for example, for the Degasperis-Procesi equation m t + (um) x + 2u x m = 0, m = u − u xx , (1.5) which is also an integrable, CH-type equation with quadratic nonlinearity. On the other hand, for other CH-type equations, in particular, for those with cubic nonlinearity, the situation is different: while considering the equation on a nonzero background again leads to problems supporting smooth solitons, changing variables (leading to zero background) results in an equation having different form, which is not equivalent to adding just a linear dispersion term; see, e.g., the case of the Novikov equation [6].…”

Section: Introductionmentioning

confidence: 99%

A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions

Monvel

Karpenko

Shepelsky

2020

Journal of Mathematical Physics

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The paper aims at developing the Riemann-Hilbert problem approach to the modified Camassa-Holm (mCH) equation in the case when the solution is assumed to approach a non-zero constant at the both infinities of the space variable. In this case, the spectral problem for the associated Lax pair equation has a continuous spectrum, which allows formulating the inverse spectral problem as a Riemann-Hilbert factorization problem with jump conditions across the real axis. We obtain a representation for the solution of the Cauchy problem for the mCH equation and also a description of certain soliton-type solutions, both regular and non-regular.

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Section: Introductionmentioning

confidence: 99%

A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions

Monvel

Karpenko

Shepelsky

2020

Journal of Mathematical Physics

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“…In 2012, Lenells first extended the Fokas unified transform method to the IBV problem for the 3 × 3 matrix Lax pair [16,17]. After that, more and more researchers begin to pay attention to studying IBV problems for integrable evolution equations with higher order Lax pairs on the half-line or on the interval, the IBV problem for the many integrable equations with 3 × 3 or 4 × 4 Lax pairs are studied, such as, the Degasperis-Procesi equation [17,18], the Sasa-Satsuma equation [19], the three wave equation [20], the coupled NLS equation [21], the vector modified KdV equation [22], the Novikov equation [23], the general coupled NLS equation [24]. the integrable spin-1 Gross-Pitaevskii equations with a 4 × 4 Lax pair [25].…”

Section: Introductionmentioning

confidence: 99%

A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line

Xia

2019

Complex Variables and Elliptic Equations

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In this paper, we consider the initial-boundary value problem of the Kundu-Eckhaus equation on the half-line by using of the Fokas unified transform method. Assuming that the solution u(x, t) exists, we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions are not independent and satisfy the so-called global relation.Assume that the solution u(x; t) of the KE equation exists, and the initial-boundary values data are defined as follows( 1.2) We will show that u(x, t) can be expressed in terms of the unique solution of a matrix RHP formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions a(λ), b(λ) and A(λ), B(λ), which obtained from the initial data u 0 (x) = u(x, 0) and the boundary data g 0 (t) = u(0, t), g 1 (t) = u x (0, t), respectively. The problem has the jump across {Imλ 2 = 0}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation, which is an algebraic equation coupling a(λ), b(λ) and A(λ), B(λ). Where a different RHP was formulated and a different representation of the solution of the KE equation was given, which are more convenient for studying the long-time asymptotic behavior for the solutions of the KE equation with the decay initial and boundary values lie in the Schwartz class.Organization of this paper is as follows. In section 2, some summary results and the basic RHP of the KE equation are given. In section 3, the spectral functions a(λ), b(λ) and A(λ), B(λ) are investigated and the RHP is presented. The last section is devoted to conclusions and discussions.

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“…Specially, one also analyzed the asymptotic behavior of the solution based on this RH problem and by employing the nonlinear version of the steepest descent method introduced by Deift and Zhou. 12 Among those, Lenells, [13][14][15] Boutet de Monvel, [16][17][18] and Xu and Fan [19][20][21] have made a great contribution. In 2012, Lenells 14 applied the unified transform approach to analyze IBV problem for integrable evolution equations whose Lax pair involving 3 × 3 matrices.…”

Section: Introductionmentioning

confidence: 99%

“…After that, many researchers began to pay attention to studying IBV problem for integrable evolution equations with higher order Lax pairs, such as the Ostrovsky-Vakhnenko equation, 17 the coupled nonlinear Schrödinger (NLS) equation, 22 vector modified KdV equation, 23 the spin-1 Gross-Pitaevskii equations, 24 and others. 18,21,25,26 These authors have also done some work on integrable equations with 2 × 2 or higher order Lax pairs. 10,[27][28][29] As is well known, many important PDEs can be used to describe nonlinear physical phenomena in nature.…”

Section: Introductionmentioning

confidence: 99%

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

Xia

2018

Math Methods in App Sciences

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In this work, we discuss the coupled modified nonlinear Schrödinger (CMNLS) equation, which describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. By use of the Fokas approach, the initial‐boundary value problem for the CMNLS equation related to a 3×3 matrix Lax pair on the half‐line is to be analyzed. Assuming that the solution {u(x,t),v(x,t)} of CMNLS equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann‐Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions s(λ) and S(λ) are not independent of each other but meet a global relationship.

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A Riemann-Hilbert Approach for the Novikov Equation

Cited by 21 publications

References 46 publications

A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions

A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions

A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

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