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Travelling wave solutions of equations in the Burgers Hierarchy
Amitava Choudhuri, Modhan Mohan Panja, Supriya Chatterjee, Benoy Talukdar
https://arxiv.org/abs/2511.06333 https://arxiv.org/pdf/2511.06333 https://arxiv.org/html/2511.06333
arXiv:2511.06333v1 Announce Type: new
Abstract: We emphasize that construction of travelling wave solutions for partial differential equations is a problem of considerable interest and thus introduce a simple algebraic method to generate such solutions for equations in the Burgers hierarchy. Our method based on a judicious use of the well known Cole-Hopf transformation is found to work satisfactorily for higher Burgers equations for which the direct method of integration is inapplicable. For Burgers equation we clearly demonstrate how does the diffusion term in the equation counteract the nonlinearity to result in a smooth wave. We envisage a similar study for higher equations in the Buggers hierarchy and establish that (i) as opposed to the solution of the Burgers equation, the purely nonlinear terms of these equations support smooth solutions and more interestingly (ii) the complete solutions of all higher-order equations are identical.
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Replaced article(s) found for nlin.SI. https://arxiv.org/list/nlin.SI/new
[1/1]:
- On differential equations invariant under a projective transformation group: integrability and re...
Marianna Euler, Norbert Euler, Francesco Oliveri
https://arxiv.org/abs/2505.09800 https://mastoxiv.page/@arXiv_nlinSI_bot/114516342516898544
- Exact solutions of the reverse space-time higher-order modified self-steepening nonlinear Schr\"o...
Yanan Wang, Xi-hu Wu
https://arxiv.org/abs/2511.03118 https://mastoxiv.page/@arXiv_nlinSI_bot/115501790094140243
- Isochronous waveforms of Li\'enard equations via commutative factorization
G. Gonzalez, O. Cornejo-Perez, J. de la Cruz, H. C. Rosu
https://arxiv.org/abs/2404.08659 https://mastoxiv.page/@arXiv_mathCA_bot/112279695996418481
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Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations
Liming Ling, Huajie Su
https://arxiv.org/abs/2510.12129 https://arxi…

Nonlinear stability of vector multi-solitons in coupled NLS and modified KdV equations
We prove that the $N$-solitons, including breathers and multi-hump solitons, of the coupled nonlinear Schrödinger (CNLS) equations are nonlinearly stable in the Sobolev space $H^{N}$. Moreover, $(N_{1},N_{2})$-solitons of the coupled modified Korteweg--de Vries (CmKdV) equations are shown to be nonlinearly stable in the Sobolev space $H^{2N_{1}+N_{2}}$. The number of negative eigenvalues of the second variation of the Lyapunov functional is $N$ for $N$-solitons of the CNLS equations, and $N_{1…

Residual Symmetry Reductions and Painlev\'e Solitons
Yan Li, Ya-Rong Xia, Ruo-Xia Yao, S. Y. Lou
https://arxiv.org/abs/2511.09077 https://arxiv.org/pdf/2511.09077 https://arxiv.org/html/2511.09077
arXiv:2511.09077v1 Announce Type: new
Abstract: This letter introduces the novel concept of Painlev\'e solitons -- waves arising from the interaction between Painlev\'e waves and solitons in integrable systems. Painlev\'e solitons may also be viewed as solitons propagating against a Painlev\'e wave background, in analogy with the established notion of elliptic solitons, which refer to solitons on an elliptic wave background. By employing a novel symmetry decomposition method aided by nonlocal residual symmetries, we explicitly construct (extended) Painlev\'e II solitons for the Korteweg-de Vries (KdV) equation and (extended) Painlev\'e IV solitons for the Boussinesq equation.
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On the integrable six-wave interaction system and its space-time shifted reduction
Mark J. Ablowitz, Ramesh Gupta, Ziad H. Musslimani, Nicholas J. Ossi
https://arxiv.org/abs/2510.10312

On the integrable six-wave interaction system and its space-time shifted reduction
The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space-time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad…

Focusing mKdV equation: Two-phase solutions and their stability analysis
Liming Ling, Xuan Sun
https://arxiv.org/abs/2510.12073 https://arxiv.org/pdf/2510.…

Focusing mKdV equation: Two-phase solutions and their stability analysis
In this work, we primarily focus on the two-phase solutions and their stability to the focusing mKdV equation. By employing the algebro-geometric approach in combination with an effective integration method, we construct explicit two-phase solutions and their corresponding wave-functions expressed in terms of the Riemann theta function. The spectral stability of two-phase solutions is examined via a modified squared-eigenfunction approach, and their stability with respect to subharmonic perturb…

Solitons in the Korteweg-de Vries Equation
G. Bueno, M. Bonehill
https://arxiv.org/abs/2510.07207 https://arxiv.org/pdf/2510.07207

Solitons in the Korteweg-de Vries Equation
We propose a numerical solution to the Korteweg-de Vries (KdV) equation using a Crank-Nicolson scheme, and compare its performance to the Fast Fourier Transform method. The properties and interactions of soliton solutions are further examined. Initial conditions were varied to analyse soliton formation in the resulting system. Performing an L$^2$ error analysis demonstrated consistency between numerical methods of solving the KdV equation and analytical solutions.

Crosslisted article(s) found for nlin.SI. https://arxiv.org/list/nlin.SI/new
[1/1]:
- Anti-commuting Solutions of the Yang-Baxter-like Matrix Equation
Mohammed Ahmed Adam Abdalrahman, Huijian Zhu, Jiu Ding, Qianglian Huang
https://arxiv.org/abs/2511.05088 https://mastoxiv.page/@arXiv_mathNA_bot/115524577525797883
- Exactly solvable Stuart-Landau models in arbitrary dimensions
Pragjyotish Bhuyan Gogoi, Rahul Ghosh, Debashis Ghoshal, Awadhesh Prasad, Ram Ramaswamy
https://arxiv.org/abs/2511.05160 https://mastoxiv.page/@arXiv_nlinCD_bot/115524434020594392
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Soliton,breathers,positons and rogue waves for the vector complex modified Korteweg-de Vries equation
Yihang Liu, Yongshuai Zhang, Maohua Li
https://arxiv.org/abs/2510.03062 htt…

Soliton,breathers,positons and rogue waves for the vector complex modified Korteweg-de Vries equation
This paper constructs the $N$-fold Darboux transformation (DT) for the vector complex modified Korteweg-de Vries (vcmKdV) equation and presents its determinant representation. Utilizing the DT and multi-fold eigenvalue degeneracy, we derive globally bounded solutions for the vcmKdV equation, including $N$-bright-bright-bright solitons, $N$-dark-bright-bright solitons, $N$-breathers, $N$-positon solutions, and $N$th-order rogue wave solutions." All these solutions are globally bounded. Graphical…

Replaced article(s) found for nlin.SI. https://arxiv.org/list/nlin.SI/new
[1/1]:
- Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument
Vladimir Dorodnitsyn, Roman Kozlov, Sergey Meleshko
https://arxiv.org/abs/2510.25391 https://mastoxiv.page/@arXiv_nlinSI_bot/115462039736354475
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